Аналіз I, Розділ 6.7: Піднесення до дійсного степеня, частина II

Я (прим. перекл. Терренс Тао) намагався зробити переклад якомога точнішим перефразуванням оригінального тексту. Коли є вибір між більш ідіоматичним підходом Lean та більш точним перекладом, я зазвичай обирав останній. Зокрема, будуть місця, де код Lean можна було б "підправити", щоб зробити його більш елегантним та ідіоматичним, але я свідомо уникав цього вибору.

Основні конструкції та результати цього розділу:

Піднесення до дійсного степеня.

Оскільки дійсні числа з Розділу 5 застаріли на користь дійсних чисел із Mathlib, а операція піднесення до степеня в Mathlib визначена без попереднього введення раціональних степенів, ми приймемо дещо незручну компромісну позицію: спочатку використовуватимемо операцію піднесення з Mathlib (та її API), коли показник є раціональним, а цим скористаємось для визначення піднесення до дійсного степеня, і в епілозі до цього розділу ми покажемо, що це збігається з операцією Mathlib.

namespace Chapter6open Sequence Real

Лема 6.7.1 (Неперервність піднесення до степеня)

lemma declaration uses 'sorry'ratPow_continuous {x α:ℝ} (hx: x > 0) {q: ℕ → ℚ}
 (hq: ((fun n ↦ (q n:ℝ)):Sequence).TendsTo α) :
 ((fun n ↦ x^(q n:ℝ)):Sequence).Convergent := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo α⊢ (↑fun n => x ^ ↑(q n)).Convergent
  -- Доказ дещо переставлено порівняно з оригінальним текстом.
  choose M hM hbound using bounded_of_convergent ⟨ α, hq ⟩x:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy M⊢ (↑fun n => x ^ ↑(q n)).Convergent
  obtain h | rfl | h := lt_trichotomy x 1inlx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:x < 1⊢ (↑fun n => x ^ ↑(q n)).Convergentinr.inlα:ℝq:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mhx:1 > 0⊢ (↑fun n => 1 ^ ↑(q n)).Convergentinr.inrx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < x⊢ (↑fun n => x ^ ↑(q n)).Convergent
  .inlx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:x < 1⊢ (↑fun n => x ^ ↑(q n)).Convergent sorryAll goals completed! 🐙
  .inr.inlα:ℝq:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mhx:1 > 0⊢ (↑fun n => 1 ^ ↑(q n)).Convergent simpinr.inlα:ℝq:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mhx:1 > 0⊢ (↑fun n => 1).Convergent; exact ⟨ 1, lim_of_const 1 ⟩All goals completed! 🐙
  have h': 1 ≤ x := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo α⊢ (↑fun n => x ^ ↑(q n)).Convergent linarithAll goals completed! 🐙
  rw [←Cauchy_iff_convergent]inr.inrx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347⊢ (↑fun n => x ^ ↑(q n)).IsCauchy
  intro ε hεinr.inrx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0⊢ ε.EventuallySteady ↑fun n => x ^ ↑(q n)
  choose K hK hclose using lim_of_roots hx (ε*x^(-M)) (byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0⊢ ε * x ^ (-M) > 0 positivityAll goals completed! 🐙)
  choose N hN hq using IsCauchy.convergent ⟨ α, hq ⟩ (1/(K+1:ℝ)) (byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0K:ℤhK:K ≥ (↑fun n => x ^ (1 / (↑n + 1))).mhclose:(ε * x ^ (-M)).CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from K) 1⊢ 1 / (↑K + 1) > 0 positivityAll goals completed! 🐙)
  simp [CloseSeq, dist_eq] at hclose hK hNinr.inrx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0K:ℤN:ℤhq:(1 / (↑K + 1)).Steady ((↑fun n => ↑(q n)).from N)hclose:∀ (n : ℤ), 0 ≤ n → K ≤ n → |(if 0 ≤ n ∧ K ≤ n then if 0 ≤ n then x ^ (↑n.toNat + 1)⁻¹ else 0 else 0) - 1| ≤ ε * x ^ (-M)hK:0 ≤ KhN:0 ≤ N⊢ ε.EventuallySteady ↑fun n => x ^ ↑(q n)
  lift N to ℕ using hNinr.inr.introx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0K:ℤhclose:∀ (n : ℤ), 0 ≤ n → K ≤ n → |(if 0 ≤ n ∧ K ≤ n then if 0 ≤ n then x ^ (↑n.toNat + 1)⁻¹ else 0 else 0) - 1| ≤ ε * x ^ (-M)hK:0 ≤ KN:ℕhq:(1 / (↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)⊢ ε.EventuallySteady ↑fun n => x ^ ↑(q n)
  lift K to ℕ using hKinr.inr.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:∀ (n : ℤ),
  0 ≤ n → ↑K ≤ n → |(if 0 ≤ n ∧ ↑K ≤ n then if 0 ≤ n then x ^ (↑n.toNat + 1)⁻¹ else 0 else 0) - 1| ≤ ε * x ^ (-M)hq:(1 / (↑↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)⊢ ε.EventuallySteady ↑fun n => x ^ ↑(q n)
  specialize hclose K (byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:∀ (n : ℤ),
  0 ≤ n → ↑K ≤ n → |(if 0 ≤ n ∧ ↑K ≤ n then if 0 ≤ n then x ^ (↑n.toNat + 1)⁻¹ else 0 else 0) - 1| ≤ ε * x ^ (-M)hq:(1 / (↑↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)⊢ 0 ≤ ↑K simpAll goals completed! 🐙) (byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:∀ (n : ℤ),
  0 ≤ n → ↑K ≤ n → |(if 0 ≤ n ∧ ↑K ≤ n then if 0 ≤ n then x ^ (↑n.toNat + 1)⁻¹ else 0 else 0) - 1| ≤ ε * x ^ (-M)hq:(1 / (↑↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)⊢ ↑K ≤ ↑K simpAll goals completed! 🐙); simp at hcloseinr.inr.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhq:(1 / (↑↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)hclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)⊢ ε.EventuallySteady ↑fun n => x ^ ↑(q n)
  use N, byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhq:(1 / (↑↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)hclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)⊢ ↑N ≥ (↑fun n => x ^ ↑(q n)).m simpAll goals completed! 🐙
  intro n hn m hmrightx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhq:(1 / (↑↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)hclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℤhn:n ≥ ((↑fun n => x ^ ↑(q n)).from ↑N).mm:ℤhm:m ≥ ((↑fun n => x ^ ↑(q n)).from ↑N).m⊢ ε.Close (((↑fun n => x ^ ↑(q n)).from ↑N).seq n) (((↑fun n => x ^ ↑(q n)).from ↑N).seq m); simp at hn hmrightx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhq:(1 / (↑↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)hclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℤm:ℤhn:↑N ≤ nhm:↑N ≤ m⊢ ε.Close (((↑fun n => x ^ ↑(q n)).from ↑N).seq n) (((↑fun n => x ^ ↑(q n)).from ↑N).seq m)
  specialize hq n (byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhq:(1 / (↑↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)hclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℤm:ℤhn:↑N ≤ nhm:↑N ≤ m⊢ n ≥ ((↑fun n => ↑(q n)).from ↑N).m simp [hn]All goals completed! 🐙) m (byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhq:(1 / (↑↑K + 1)).Steady ((↑fun n => ↑(q n)).from ↑N)hclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℤm:ℤhn:↑N ≤ nhm:↑N ≤ m⊢ m ≥ ((↑fun n => ↑(q n)).from ↑N).m simp [hm]All goals completed! 🐙)
  simp [Close, hn, hm, dist_eq] at hq ⊢rightx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℤm:ℤhn:↑N ≤ nhm:↑N ≤ mhq:|(if 0 ≤ n then ↑(q n.toNat) else 0) - if 0 ≤ m then ↑(q m.toNat) else 0| ≤ (↑K + 1)⁻¹⊢ |(if 0 ≤ n then x ^ ↑(q n.toNat) else 0) - if 0 ≤ m then x ^ ↑(q m.toNat) else 0| ≤ ε
  have : 0 ≤ (N:ℤ) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo α⊢ (↑fun n => x ^ ↑(q n)).Convergent simpAll goals completed! 🐙
  lift n to ℕ using byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℤm:ℤhn:↑N ≤ nhm:↑N ≤ mhq:|(if 0 ≤ n then ↑(q n.toNat) else 0) - if 0 ≤ m then ↑(q m.toNat) else 0| ≤ (↑K + 1)⁻¹this:0 ≤ ↑_fvar.28515 := of_eq_true (Nat.cast_nonneg._simp_1 _fvar.28515)⊢ 0 ≤ n linarithAll goals completed! 🐙
  lift m to ℕ using byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)m:ℤhm:↑N ≤ mthis:0 ≤ ↑_fvar.28515 := of_eq_true (Nat.cast_nonneg._simp_1 _fvar.28515)n:ℕhn:↑N ≤ ↑nhq:|(if 0 ≤ ↑n then ↑(q (↑n).toNat) else 0) - if 0 ≤ m then ↑(q m.toNat) else 0| ≤ (↑K + 1)⁻¹⊢ 0 ≤ m linarithAll goals completed! 🐙
  simp at hn hm hq ⊢right.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)this:0 ≤ ↑_fvar.28515 := ?_mvar.57455n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹⊢ |x ^ ↑(q n) - x ^ ↑(q m)| ≤ ε
  obtain hqq | hqq := le_or_gt (q m) (q n)right.intro.intro.inlx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)this:0 ≤ ↑_fvar.28515 := ?_mvar.57455n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q n⊢ |x ^ ↑(q n) - x ^ ↑(q m)| ≤ εright.intro.intro.inrx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)this:0 ≤ ↑_fvar.28515 := ?_mvar.57455n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q m⊢ |x ^ ↑(q n) - x ^ ↑(q m)| ≤ ε
  .right.intro.intro.inlx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)this:0 ≤ ↑_fvar.28515 := ?_mvar.57455n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q n⊢ |x ^ ↑(q n) - x ^ ↑(q m)| ≤ ε replace : x^(q m:ℝ) ≤ x^(q n:ℝ) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo α⊢ (↑fun n => x ^ ↑(q n)).Convergent rw [rpow_le_rpow_left_iff h]x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)this:0 ≤ ↑_fvar.28515 := ?_mvar.57455n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q n⊢ ↑(q m) ≤ ↑(q n); norm_castAll goals completed! 🐙
    rw [abs_of_nonneg (by linarith)]right.intro.intro.inlx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := ?_mvar.72180⊢ x ^ ↑(q n) - x ^ ↑(q m) ≤ ε
    calc
      _ = x^(q m:ℝ) * (x^(q n - q m:ℝ) - 1) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ x ^ ↑(q n) - x ^ ↑(q m) = x ^ ↑(q m) * (x ^ (↑(q n) - ↑(q m)) - 1) ring_nfx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ x ^ ↑(q n) - x ^ ↑(q m) = -x ^ ↑(q m) + x ^ ↑(q m) * x ^ (↑(q n) - ↑(q m)); rw [←rpow_add (by linarith)]x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ x ^ ↑(q n) - x ^ ↑(q m) = -x ^ ↑(q m) + x ^ (↑(q m) + (↑(q n) - ↑(q m))); ring_nfAll goals completed! 🐙
      _ ≤ x^M * (x^(1/(K+1:ℝ)) - 1) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ x ^ ↑(q m) * (x ^ (↑(q n) - ↑(q m)) - 1) ≤ x ^ M * (x ^ (1 / (↑K + 1)) - 1)
        gcongrc0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 0 ≤ x ^ (↑(q n) - ↑(q m)) - 1h₁.hxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 1 ≤ xh₁.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ ↑(q m) ≤ Mh₂.h.hxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 1 ≤ xh₂.h.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ ↑(q n) - ↑(q m) ≤ 1 / (↑K + 1) <;>c0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 0 ≤ x ^ (↑(q n) - ↑(q m)) - 1h₁.hxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 1 ≤ xh₁.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ ↑(q m) ≤ Mh₂.h.hxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 1 ≤ xh₂.h.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ ↑(q n) - ↑(q m) ≤ 1 / (↑K + 1) try exact h'h₂.h.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ ↑(q n) - ↑(q m) ≤ 1 / (↑K + 1)
        .c0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 0 ≤ x ^ (↑(q n) - ↑(q m)) - 1 rw [sub_nonneg]c0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 1 ≤ x ^ (↑(q n) - ↑(q m)); apply one_le_rpow h'c0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 0 ≤ ↑(q n) - ↑(q m); norm_castc0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 0 ≤ q n - q m; linarithAll goals completed! 🐙
        .h₁.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ ↑(q m) ≤ M specialize hbound mh₁.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0h:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)hbound:|(↑fun n => ↑(q n)).seq ↑m| ≤ M⊢ ↑(q m) ≤ M; simp_all [abs_le']All goals completed! 🐙
        grind [abs_le']All goals completed! 🐙
      _ ≤ x^M * (ε * x^(-M)) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ x ^ M * (x ^ (1 / (↑K + 1)) - 1) ≤ x ^ M * (ε * x ^ (-M)) gcongrhx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ x ^ (1 / (↑K + 1)) - 1 ≤ ε * x ^ (-M); grind [abs_le']All goals completed! 🐙
      _ = ε := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ x ^ M * (ε * x ^ (-M)) = ε rw [mul_comm, mul_assoc, ←rpow_add]x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ ε * x ^ (-M + M) = εhxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 0 < x; simphxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q m ≤ q nthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.62634) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.60427) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1) _fvar.72051)⊢ 0 < x; linarithAll goals completed! 🐙
  replace : x^(q n:ℝ) ≤ x^(q m:ℝ) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo α⊢ (↑fun n => x ^ ↑(q n)).Convergent rw [rpow_le_rpow_left_iff h]x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)this:0 ≤ ↑_fvar.28515 := ?_mvar.57455n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q m⊢ ↑(q n) ≤ ↑(q m); norm_castx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)this:0 ≤ ↑_fvar.28515 := ?_mvar.57455n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q m⊢ q n ≤ q m; linarithAll goals completed! 🐙
  rw [abs_of_nonpos (by linarith)]right.intro.intro.inrx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := ?_mvar.1347ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := ?_mvar.147519⊢ -(x ^ ↑(q n) - x ^ ↑(q m)) ≤ ε
  calc
    _ = x^(q n:ℝ) * (x^(q m - q n:ℝ) - 1) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ -(x ^ ↑(q n) - x ^ ↑(q m)) = x ^ ↑(q n) * (x ^ (↑(q m) - ↑(q n)) - 1) ring_nfx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ -x ^ ↑(q n) + x ^ ↑(q m) = -x ^ ↑(q n) + x ^ ↑(q n) * x ^ (↑(q m) - ↑(q n)); rw [←rpow_add]x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ -x ^ ↑(q n) + x ^ ↑(q m) = -x ^ ↑(q n) + x ^ (↑(q n) + (↑(q m) - ↑(q n)))hxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 0 < x; ring_nfhxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 0 < x; positivityAll goals completed! 🐙
    _ ≤ x^M * (x^(1/(K+1:ℝ)) - 1) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ x ^ ↑(q n) * (x ^ (↑(q m) - ↑(q n)) - 1) ≤ x ^ M * (x ^ (1 / (↑K + 1)) - 1)
      gcongrc0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 0 ≤ x ^ (↑(q m) - ↑(q n)) - 1h₁.hxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 1 ≤ xh₁.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ ↑(q n) ≤ Mh₂.h.hxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 1 ≤ xh₂.h.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ ↑(q m) - ↑(q n) ≤ 1 / (↑K + 1) <;>c0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 0 ≤ x ^ (↑(q m) - ↑(q n)) - 1h₁.hxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 1 ≤ xh₁.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ ↑(q n) ≤ Mh₂.h.hxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 1 ≤ xh₂.h.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ ↑(q m) - ↑(q n) ≤ 1 / (↑K + 1) try exact h'h₂.h.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ ↑(q m) - ↑(q n) ≤ 1 / (↑K + 1)
      .c0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 0 ≤ x ^ (↑(q m) - ↑(q n)) - 1 rw [sub_nonneg]c0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 1 ≤ x ^ (↑(q m) - ↑(q n)); apply one_le_rpow h'c0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 0 ≤ ↑(q m) - ↑(q n); norm_castc0x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 0 ≤ q m - q n; linarithAll goals completed! 🐙
      .h₁.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ ↑(q n) ≤ M specialize hbound nh₁.hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0h:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))hbound:|(↑fun n => ↑(q n)).seq ↑n| ≤ M⊢ ↑(q n) ≤ M; simp_all [abs_le']All goals completed! 🐙
      grind [abs_le']All goals completed! 🐙
    _ ≤ x^M * (ε * x^(-M)) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ x ^ M * (x ^ (1 / (↑K + 1)) - 1) ≤ x ^ M * (ε * x ^ (-M)) gcongrhx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ x ^ (1 / (↑K + 1)) - 1 ≤ ε * x ^ (-M); simp_all [abs_le']All goals completed! 🐙
    _ = ε := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ x ^ M * (ε * x ^ (-M)) = ε rw [mul_comm, mul_assoc, ←rpow_add]x:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ ε * x ^ (-M + M) = εhxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 0 < x; simphxx:ℝα:ℝhx:x > 0q:ℕ → ℚhq✝:(↑fun n => ↑(q n)).TendsTo αM:ℝhM:M ≥ 0hbound:(↑fun n => ↑(q n)).BoundedBy Mh:1 < xh':1 ≤ _fvar.321 := 
  le_of_not_gt fun a =>
    Mathlib.Tactic.Linarith.lt_irrefl
      (Eq.mp
        (congrArg (fun _a => _a < 0)
          (Mathlib.Tactic.Ring.of_eq
            (Mathlib.Tactic.Ring.add_congr
              (Mathlib.Tactic.Ring.sub_congr
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_mul _fvar.321 (Nat.rawCast 1)
                      (Mathlib.Tactic.Ring.neg_one_mul
                        (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                          (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                            (Eq.refl (Int.negOfNat 1))))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 1)
                    (Mathlib.Tactic.Ring.add_pf_zero_add (_fvar.321 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
              (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf _fvar.321)
                (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one))
                (Mathlib.Tactic.Ring.sub_pf
                  (Mathlib.Tactic.Ring.neg_add
                    (Mathlib.Tactic.Ring.neg_one_mul
                      (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                        (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                          (Eq.refl (Int.negOfNat 1)))))
                    Mathlib.Tactic.Ring.neg_zero)
                  (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 1).rawCast
                    (Mathlib.Tactic.Ring.add_pf_add_zero (_fvar.321 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
              (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                (Mathlib.Meta.NormNum.IsInt.to_isNat
                  (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                    (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                    (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0))))
                (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                  (Mathlib.Tactic.Ring.add_overlap_pf_zero _fvar.321 (Nat.rawCast 1)
                    (Mathlib.Meta.NormNum.IsInt.to_isNat
                      (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                        (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1))
                        (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1))
                        (Eq.refl (Int.ofNat 0)))))
                  (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
            (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero))))
        (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.447)
          (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))ε:ℝhε:ε > 0N:ℕK:ℕhclose:|x ^ (↑K + 1)⁻¹ - 1| ≤ ε * x ^ (-M)n:ℕm:ℕhn:N ≤ nhm:N ≤ mhq:|↑(q n) - ↑(q m)| ≤ (↑K + 1)⁻¹hqq:q n < q mthis:_fvar.321 ^ ↑(@_fvar.324 _fvar.60427) ≤ _fvar.321 ^ ↑(@_fvar.324 _fvar.62634) := 
  Eq.mpr (id (congrArg (fun _a => _a) (propext (Real.rpow_le_rpow_left_iff _fvar.447))))
    (Eq.mpr (id Rat.cast_le._simp_1)
      (le_of_not_gt fun a =>
        Mathlib.Tactic.Linarith.lt_irrefl
          (Eq.mp
            (congrArg (fun _a => _a < 0)
              (Mathlib.Tactic.Ring.of_eq
                (Mathlib.Tactic.Ring.add_congr
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_lt (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * Nat.rawCast 1)
                        (Mathlib.Tactic.Ring.add_pf_zero_add
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)))))
                  (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.62634))
                    (Mathlib.Tactic.Ring.atom_pf (@_fvar.324 _fvar.60427))
                    (Mathlib.Tactic.Ring.sub_pf
                      (Mathlib.Tactic.Ring.neg_add
                        (Mathlib.Tactic.Ring.neg_mul (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                          (Mathlib.Tactic.Ring.neg_one_mul
                            (Mathlib.Meta.NormNum.IsInt.to_raw_eq
                              (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul)
                                (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                                (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                                (Eq.refl (Int.negOfNat 1))))))
                        Mathlib.Tactic.Ring.neg_zero)
                      (Mathlib.Tactic.Ring.add_pf_add_gt
                        (@_fvar.324 _fvar.60427 ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast)
                        (Mathlib.Tactic.Ring.add_pf_add_zero
                          (@_fvar.324 _fvar.62634 ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))))
                  (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                    (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.60427) (Nat.rawCast 1)
                      (Mathlib.Meta.NormNum.IsInt.to_isNat
                        (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                          (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                          (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))))
                    (Mathlib.Tactic.Ring.add_pf_add_overlap_zero
                      (Mathlib.Tactic.Ring.add_overlap_pf_zero (@_fvar.324 _fvar.62634) (Nat.rawCast 1)
                        (Mathlib.Meta.NormNum.IsInt.to_isNat
                          (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd)
                            (Mathlib.Meta.NormNum.IsInt.of_raw ℚ (Int.negOfNat 1))
                            (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℚ 1))
                            (Eq.refl (Int.ofNat 0)))))
                      (Mathlib.Tactic.Ring.add_pf_zero_add 0))))
                (Mathlib.Tactic.Ring.cast_zero (Mathlib.Meta.NormNum.isNat_ofNat ℚ (Eq.refl 0)))))
            (Mathlib.Tactic.Linarith.add_neg (Mathlib.Tactic.Linarith.sub_neg_of_lt _fvar.72056)
              (Mathlib.Tactic.Linarith.sub_neg_of_lt a)))))⊢ 0 < x; positivityAll goals completed! 🐙

lemma declaration uses 'sorry'ratPow_lim_uniq {x α:ℝ} (hx: x > 0) {q q': ℕ → ℚ}
 (hq: ((fun n ↦ (q n:ℝ)):Sequence).TendsTo α)
 (hq': ((fun n ↦ (q' n:ℝ)):Sequence).TendsTo α) :
 lim ((fun n ↦ x^(q n:ℝ)):Sequence) = lim ((fun n ↦ x^(q' n:ℝ)):Sequence) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo α⊢ (lim ↑fun n => x ^ ↑(q n)) = lim ↑fun n => x ^ ↑(q' n)
 -- Доведення написане так, щоб відповідати структурі оригінального тексту.
  set r := q - q'x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253⊢ (lim ↑fun n => x ^ ↑(q n)) = lim ↑fun n => x ^ ↑(q' n)
  suffices: (fun n ↦ x^(r n:ℝ):Sequence).TendsTo 1x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).TendsTo 1⊢ (lim ↑fun n => x ^ ↑(q n)) = lim ↑fun n => x ^ ↑(q' n)thisx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253⊢ (↑fun n => x ^ ↑(r n)).TendsTo 1
  .x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).TendsTo 1⊢ (lim ↑fun n => x ^ ↑(q n)) = lim ↑fun n => x ^ ↑(q' n) rw [←mul_one (lim ((fun n ↦ x^(q' n:ℝ)):Sequence))]x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).TendsTo 1⊢ (lim ↑fun n => x ^ ↑(q n)) = (lim ↑fun n => x ^ ↑(q' n)) * 1
    rw [lim_eq] at thisx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).Convergent ∧ (lim ↑fun n => x ^ ↑(r n)) = 1⊢ (lim ↑fun n => x ^ ↑(q n)) = (lim ↑fun n => x ^ ↑(q' n)) * 1
    convert (lim_mul (b := (fun n ↦ x^(r n:ℝ):Sequence)) (ratPow_continuous hx hq') this.1).2h.e'_2.h.e'_1x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).Convergent ∧ (lim ↑fun n => x ^ ↑(r n)) = 1⊢ (↑fun n => x ^ ↑(q n)) = (↑fun n => x ^ ↑(q' n)) * ↑fun n => x ^ ↑(r n)h.e'_3.h.e'_6x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).Convergent ∧ (lim ↑fun n => x ^ ↑(r n)) = 1⊢ 1 = lim ↑fun n => x ^ ↑(r n)
    .h.e'_2.h.e'_1x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).Convergent ∧ (lim ↑fun n => x ^ ↑(r n)) = 1⊢ (↑fun n => x ^ ↑(q n)) = (↑fun n => x ^ ↑(q' n)) * ↑fun n => x ^ ↑(r n) rw [mul_coe]h.e'_2.h.e'_1x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).Convergent ∧ (lim ↑fun n => x ^ ↑(r n)) = 1⊢ (↑fun n => x ^ ↑(q n)) = ↑fun n => x ^ ↑(q' n) * x ^ ↑(r n)
      rcongr _ nh.e'_2.h.e'_1.seq.h.e_self.e_a.hx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).Convergent ∧ (lim ↑fun n => x ^ ↑(r n)) = 1x✝:ℤn:ℕ⊢ x ^ ↑(q n) = x ^ ↑(q' n) * x ^ ↑(r n)
      rw [←rpow_add (by linarith)]h.e'_2.h.e'_1.seq.h.e_self.e_a.hx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253this:(↑fun n => x ^ ↑(r n)).Convergent ∧ (lim ↑fun n => x ^ ↑(r n)) = 1x✝:ℤn:ℕ⊢ x ^ ↑(q n) = x ^ (↑(q' n) + ↑(r n))
      simp [r]All goals completed! 🐙
    grindAll goals completed! 🐙
  intro ε hεthisx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  have h1 := lim_of_roots hxthisx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  have h2 := tendsTo_inv h1 (byx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1))).TendsTo 1 := Chapter6.Sequence.lim_of_roots _fvar.236251⊢ 1 ≠ 0 norm_numAll goals completed! 🐙)
  choose K1 hK1 h3 using h1 ε hεthisx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℤhK1:K1 ≥ (↑fun n => x ^ (1 / (↑n + 1))).mh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from K1) 1⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  choose K2 hK2 h4 using h2 ε hεthisx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℤhK1:K1 ≥ (↑fun n => x ^ (1 / (↑n + 1))).mh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from K1) 1K2:ℤhK2:K2 ≥ (↑fun n => x ^ (1 / (↑n + 1)))⁻¹.mh4:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1)))⁻¹.from K2) 1⁻¹⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  simp [Inv.inv] at hK1 hK2thisx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℤh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from K1) 1K2:ℤh4:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1)))⁻¹.from K2) 1⁻¹hK1:0 ≤ K1hK2:0 ≤ K2⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  lift K1 to ℕ using hK1this.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K2:ℤh4:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1)))⁻¹.from K2) 1⁻¹hK2:0 ≤ K2K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1; lift K2 to ℕ using hK2this.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1)))⁻¹.from ↑K2) 1⁻¹⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  simp [inv_coe] at h4this.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  set K := max K1 K2this.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1K:ℕ := max _fvar.255480 _fvar.255540⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  have hr := tendsTo_sub hq hq'this.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1K:ℕ := max _fvar.255480 _fvar.255540hr:?_mvar.257014 := Chapter6.Sequence.tendsTo_sub _fvar.236254 _fvar.236255⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  rw [sub_coe] at hrthis.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1K:ℕ := max _fvar.255480 _fvar.255540hr:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)⊢ ε.EventuallyClose (↑fun n => x ^ ↑(r n)) 1
  choose N hN hr using hr (1 / (K + 1:ℝ)) (byx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1))).TendsTo 1 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1)))⁻¹.TendsTo 1⁻¹ := 
  Chapter6.Sequence.tendsTo_inv _fvar.250363
    (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)
      (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Eq.refl false))K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1K:ℕ := max _fvar.255480 _fvar.255540hr:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)⊢ 1 / (↑K + 1) > 0 positivityAll goals completed! 🐙)
  refine ⟨ N, byx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1))).TendsTo 1 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1)))⁻¹.TendsTo 1⁻¹ := 
  Chapter6.Sequence.tendsTo_inv _fvar.250363
    (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)
      (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Eq.refl false))K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1K:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mhr:(1 / (↑K + 1)).CloseSeq ((↑fun n => ↑(q n) - ↑(q' n)).from N) (α - α)⊢ N ≥ (↑fun n => x ^ ↑(r n)).m simp_allAll goals completed! 🐙, ?_ ⟩
  intro n hnthis.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1K:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mhr:(1 / (↑K + 1)).CloseSeq ((↑fun n => ↑(q n) - ↑(q' n)).from N) (α - α)n:ℤhn:n ≥ ((↑fun n => x ^ ↑(r n)).from N).m⊢ ε.Close (((↑fun n => x ^ ↑(r n)).from N).seq n) 1; simp at hnthis.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1K:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mhr:(1 / (↑K + 1)).CloseSeq ((↑fun n => ↑(q n) - ↑(q' n)).from N) (α - α)n:ℤhn:0 ≤ n ∧ N ≤ n⊢ ε.Close (((↑fun n => x ^ ↑(r n)).from N).seq n) 1
  specialize h3 K (byx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1))).TendsTo 1 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1)))⁻¹.TendsTo 1⁻¹ := 
  Chapter6.Sequence.tendsTo_inv _fvar.250363
    (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)
      (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Eq.refl false))K1:ℕh3:ε.CloseSeq ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1) 1K2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1K:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mhr:(1 / (↑K + 1)).CloseSeq ((↑fun n => ↑(q n) - ↑(q' n)).from N) (α - α)n:ℤhn:0 ≤ n ∧ N ≤ n⊢ ↑K ≥ ((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1).m simp [K]All goals completed! 🐙); specialize h4 K (byx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1))).TendsTo 1 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1)))⁻¹.TendsTo 1⁻¹ := 
  Chapter6.Sequence.tendsTo_inv _fvar.250363
    (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)
      (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Eq.refl false))K1:ℕK2:ℕh4:ε.CloseSeq ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2) 1K:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mhr:(1 / (↑K + 1)).CloseSeq ((↑fun n => ↑(q n) - ↑(q' n)).from N) (α - α)n:ℤhn:0 ≤ n ∧ N ≤ nh3:ε.Close (((↑fun n => x ^ (1 / (↑n + 1))).from ↑K1).seq ↑K) 1⊢ ↑K ≥ ((↑fun n => (x ^ (↑n + 1)⁻¹)⁻¹).from ↑K2).m simp [K]All goals completed! 🐙)
  simp [hn, dist_eq, abs_le', K, -Nat.cast_max] at h3 h4 ⊢this.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mhr:(1 / (↑K + 1)).CloseSeq ((↑fun n => ↑(q n) - ↑(q' n)).from N) (α - α)n:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹⊢ x ^ ↑(r n.toNat) ≤ ε + 1 ∧ 1 ≤ ε + x ^ ↑(r n.toNat)
  specialize hr n (byx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1))).TendsTo 1 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:(↑fun n => _fvar.236249 ^ (1 / (↑n + 1)))⁻¹.TendsTo 1⁻¹ := 
  Chapter6.Sequence.tendsTo_inv _fvar.250363
    (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)
      (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero) (Eq.refl false))K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mhr:(1 / (↑K + 1)).CloseSeq ((↑fun n => ↑(q n) - ↑(q' n)).from N) (α - α)n:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹⊢ n ≥ ((↑fun n => ↑(q n) - ↑(q' n)).from N).m simp [hn]All goals completed! 🐙)
  simp [Close, hn, abs_le'] at hrthis.intro.introx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)⊢ x ^ ↑(r n.toNat) ≤ ε + 1 ∧ 1 ≤ ε + x ^ ↑(r n.toNat)
  obtain h | rfl | h := lt_trichotomy x 1this.intro.intro.inlx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:x < 1⊢ x ^ ↑(r n.toNat) ≤ ε + 1 ∧ 1 ≤ ε + x ^ ↑(r n.toNat)this.intro.intro.inr.inlα:ℝq:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nhr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)hx:1 > 0h1:(↑fun n => 1 ^ (1 / (↑n + 1))).TendsTo 1h2:(↑fun n => 1 ^ (1 / (↑n + 1)))⁻¹.TendsTo 1⁻¹h3:1 ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + 1 ^ (↑(max K1 K2) + 1)⁻¹h4:(1 ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (1 ^ (↑(max K1 K2) + 1)⁻¹)⁻¹⊢ 1 ^ ↑(r n.toNat) ≤ ε + 1 ∧ 1 ≤ ε + 1 ^ ↑(r n.toNat)this.intro.intro.inr.inrx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < x⊢ x ^ ↑(r n.toNat) ≤ ε + 1 ∧ 1 ≤ ε + x ^ ↑(r n.toNat)
  .this.intro.intro.inlx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:x < 1⊢ x ^ ↑(r n.toNat) ≤ ε + 1 ∧ 1 ≤ ε + x ^ ↑(r n.toNat) sorryAll goals completed! 🐙
  .this.intro.intro.inr.inlα:ℝq:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nhr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)hx:1 > 0h1:(↑fun n => 1 ^ (1 / (↑n + 1))).TendsTo 1h2:(↑fun n => 1 ^ (1 / (↑n + 1)))⁻¹.TendsTo 1⁻¹h3:1 ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + 1 ^ (↑(max K1 K2) + 1)⁻¹h4:(1 ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (1 ^ (↑(max K1 K2) + 1)⁻¹)⁻¹⊢ 1 ^ ↑(r n.toNat) ≤ ε + 1 ∧ 1 ≤ ε + 1 ^ ↑(r n.toNat) simpthis.intro.intro.inr.inlα:ℝq:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nhr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)hx:1 > 0h1:(↑fun n => 1 ^ (1 / (↑n + 1))).TendsTo 1h2:(↑fun n => 1 ^ (1 / (↑n + 1)))⁻¹.TendsTo 1⁻¹h3:1 ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + 1 ^ (↑(max K1 K2) + 1)⁻¹h4:(1 ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (1 ^ (↑(max K1 K2) + 1)⁻¹)⁻¹⊢ 0 ≤ ε; linarithAll goals completed! 🐙
  have h5 : x ^ (r n.toNat:ℝ) ≤ x^(K + 1:ℝ)⁻¹ := byx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo α⊢ (lim ↑fun n => x ^ ↑(q n)) = lim ↑fun n => x ^ ↑(q' n) gcongrhxx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < x⊢ 1 ≤ xhyzx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < x⊢ ↑(r n.toNat) ≤ (↑K + 1)⁻¹; linarithhyzx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < x⊢ ↑(r n.toNat) ≤ (↑K + 1)⁻¹; simp_all [r]All goals completed! 🐙
  have h6 : (x^(K + 1:ℝ)⁻¹)⁻¹ ≤ x ^ (r n.toNat:ℝ) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo α⊢ (lim ↑fun n => x ^ ↑(q n)) = lim ↑fun n => x ^ ↑(q' n)
    rw [←rpow_neg (by linarith)]x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < xh5:_fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) ≤ _fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹ := ?_mvar.378284⊢ x ^ (-(↑K + 1)⁻¹) ≤ x ^ ↑(r n.toNat)
    gcongrhxx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < xh5:_fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) ≤ _fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹ := ?_mvar.378284⊢ 1 ≤ xhyzx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < xh5:_fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) ≤ _fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹ := ?_mvar.378284⊢ -(↑K + 1)⁻¹ ≤ ↑(r n.toNat); linarithhyzx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < xh5:_fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) ≤ _fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹ := ?_mvar.378284⊢ -(↑K + 1)⁻¹ ≤ ↑(r n.toNat)
    simp [r]hyzx:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < xh5:_fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) ≤ _fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹ := ?_mvar.378284⊢ ↑(q' n.toNat) ≤ ↑(q n.toNat) + (↑K + 1)⁻¹; linarithAll goals completed! 🐙
  split_andsthis.intro.intro.inr.inr.refine_1x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < xh5:_fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) ≤ _fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹ := ?_mvar.378284h6:(_fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹)⁻¹ ≤ _fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) := ?_mvar.420978⊢ x ^ ↑(r n.toNat) ≤ ε + 1this.intro.intro.inr.inr.refine_2x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < xh5:_fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) ≤ _fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹ := ?_mvar.378284h6:(_fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹)⁻¹ ≤ _fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) := ?_mvar.420978⊢ 1 ≤ ε + x ^ ↑(r n.toNat) <;>this.intro.intro.inr.inr.refine_1x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < xh5:_fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) ≤ _fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹ := ?_mvar.378284h6:(_fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹)⁻¹ ≤ _fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) := ?_mvar.420978⊢ x ^ ↑(r n.toNat) ≤ ε + 1this.intro.intro.inr.inr.refine_2x:ℝα:ℝhx:x > 0q:ℕ → ℚq':ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo αhq':(↑fun n => ↑(q' n)).TendsTo αr:ℕ → ℚ := _fvar.236252 - _fvar.236253ε:ℝhε:ε > 0h1:?_mvar.250353 := Chapter6.Sequence.lim_of_roots _fvar.236251h2:?_mvar.250372 := Chapter6.Sequence.tendsTo_inv _fvar.250363 ?_mvar.250381K1:ℕK2:ℕK:ℕ := max _fvar.255480 _fvar.255540hr✝:(↑fun n => ↑(q n) - ↑(q' n)).TendsTo (α - α)N:ℤhN:N ≥ (↑fun n => ↑(q n) - ↑(q' n)).mn:ℤhn:0 ≤ n ∧ N ≤ nh3:x ^ (↑(max K1 K2) + 1)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + x ^ (↑(max K1 K2) + 1)⁻¹h4:(x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹ ≤ ε + 1 ∧ 1 ≤ ε + (x ^ (↑(max K1 K2) + 1)⁻¹)⁻¹hr:↑(q n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q' n.toNat) ∧ ↑(q' n.toNat) ≤ (↑K + 1)⁻¹ + ↑(q n.toNat)h:1 < xh5:_fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) ≤ _fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹ := ?_mvar.378284h6:(_fvar.236249 ^ (↑_fvar.256839 + 1)⁻¹)⁻¹ ≤ _fvar.236249 ^ ↑(@_fvar.236354 (Int.toNat _fvar.274816)) := ?_mvar.420978⊢ 1 ≤ ε + x ^ ↑(r n.toNat) linarithAll goals completed! 🐙

theorem Real.eq_lim_of_rat (α:ℝ) : ∃ q: ℕ → ℚ, ((fun n ↦ (q n:ℝ)):Sequence).TendsTo α := byα:ℝ⊢ ∃ q, (↑fun n => ↑(q n)).TendsTo α
  choose q hcauchy hLIM using (Chapter5.Real.equivR.symm α).eq_limα:ℝq:ℕ → ℚhcauchy:(↑q).IsCauchyhLIM:Chapter5.Real.equivR.symm α = Chapter5.LIM q⊢ ∃ q, (↑fun n => ↑(q n)).TendsTo α; use qhα:ℝq:ℕ → ℚhcauchy:(↑q).IsCauchyhLIM:Chapter5.Real.equivR.symm α = Chapter5.LIM q⊢ (↑fun n => ↑(q n)).TendsTo α
  apply lim_eq_LIM at hcauchyhα:ℝq:ℕ → ℚhLIM:Chapter5.Real.equivR.symm α = Chapter5.LIM qhcauchy:(↑↑q).TendsTo (Chapter5.Real.equivR (Chapter5.LIM q))⊢ (↑fun n => ↑(q n)).TendsTo α
  simp only [←hLIM, Equiv.apply_symm_apply] at hcauchyhα:ℝq:ℕ → ℚhLIM:Chapter5.Real.equivR.symm α = Chapter5.LIM qhcauchy:(↑↑q).TendsTo α⊢ (↑fun n => ↑(q n)).TendsTo α
  convert hcauchyh.e'_1α:ℝq:ℕ → ℚhLIM:Chapter5.Real.equivR.symm α = Chapter5.LIM qhcauchy:(↑↑q).TendsTo α⊢ (↑fun n => ↑(q n)) = ↑↑q; aesopAll goals completed! 🐙

Визначення 6.7.2 (Піднесення до дійсного показника)

noncomputable abbrev Real.rpow (x:ℝ) (α:ℝ) :ℝ := lim ((fun n ↦ x^((eq_lim_of_rat α).choose n:ℝ)):Sequence)

lemma Real.rpow_eq_lim_ratPow {x α:ℝ} (hx: x > 0) {q: ℕ → ℚ}
 (hq: ((fun n ↦ (q n:ℝ)):Sequence).TendsTo α) :
 rpow x α = lim ((fun n ↦ x^(q n:ℝ)):Sequence) :=
   ratPow_lim_uniq hx (eq_lim_of_rat α).choose_spec hq

lemma Real.ratPow_tendsto_rpow {x α:ℝ} (hx: x > 0) {q: ℕ → ℚ}
 (hq: ((fun n ↦ (q n:ℝ)):Sequence).TendsTo α) :
 ((fun n ↦ x^(q n:ℝ)):Sequence).TendsTo (rpow x α) := byx:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo α⊢ (↑fun n => x ^ ↑(q n)).TendsTo (rpow x α)
  rw [lim_eq]x:ℝα:ℝhx:x > 0q:ℕ → ℚhq:(↑fun n => ↑(q n)).TendsTo α⊢ (↑fun n => x ^ ↑(q n)).Convergent ∧ (lim ↑fun n => x ^ ↑(q n)) = rpow x α
  exact ⟨ ratPow_continuous hx hq, (rpow_eq_lim_ratPow hx hq).symm ⟩All goals completed! 🐙

lemma Real.rpow_of_rat_eq_ratPow {x:ℝ} (hx: x > 0) {q: ℚ} :
  rpow x (q:ℝ) = x^(q:ℝ) := byx:ℝhx:x > 0q:ℚ⊢ rpow x ↑q = x ^ ↑q
  convert rpow_eq_lim_ratPow hx (α := q) (lim_of_const _)h.e'_3x:ℝhx:x > 0q:ℚ⊢ x ^ ↑q = lim ↑fun n => x ^ ↑q
  exact (lim_eq.mp (lim_of_const _)).2.symmAll goals completed! 🐙

Твердження 6.7.3(a) / Вправа 6.7.1

theorem declaration uses 'sorry'Real.ratPow_nonneg {x:ℝ} (hx: x > 0) (q:ℝ) : rpow x q ≥ 0 := byx:ℝhx:x > 0q:ℝ⊢ rpow x q ≥ 0
  sorryAll goals completed! 🐙

Твердження 6.7.3(b)

theorem Real.ratPow_add {x:ℝ} (hx: x > 0) (q r:ℝ) : rpow x (q+r) = rpow x q * rpow x r := byx:ℝhx:x > 0q:ℝr:ℝ⊢ rpow x (q + r) = rpow x q * rpow x r
  choose q' hq' using eq_lim_of_rat qx:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo q⊢ rpow x (q + r) = rpow x q * rpow x r
  choose r' hr' using eq_lim_of_rat rx:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo r⊢ rpow x (q + r) = rpow x q * rpow x r
  have hq'r' := tendsTo_add hq' hr'x:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':?_mvar.482424 := Chapter6.Sequence.tendsTo_add _fvar.482409 _fvar.482418⊢ rpow x (q + r) = rpow x q * rpow x r
  rw [add_coe] at hq'r'x:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n) + ↑(r' n)).TendsTo (q + r)⊢ rpow x (q + r) = rpow x q * rpow x r
  convert_to ((fun n ↦ ((q' n + r' n:ℚ):ℝ)):Sequence).TendsTo (q + r) at hq'r'h.e'_1x:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n) + ↑(r' n)).TendsTo (q + r)⊢ (↑fun n => ↑(q' n) + ↑(r' n)) = ↑fun n => ↑(q' n + r' n)x:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n + r' n)).TendsTo (q + r)⊢ rpow x (q + r) = rpow x q * rpow x r
  .h.e'_1x:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n) + ↑(r' n)).TendsTo (q + r)⊢ (↑fun n => ↑(q' n) + ↑(r' n)) = ↑fun n => ↑(q' n + r' n) aesopAll goals completed! 🐙
  have h1 := ratPow_continuous hx hq'x:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n + r' n)).TendsTo (q + r)h1:?_mvar.485289 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482409⊢ rpow x (q + r) = rpow x q * rpow x r
  have h2 := ratPow_continuous hx hr'x:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n + r' n)).TendsTo (q + r)h1:?_mvar.485289 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482409h2:?_mvar.485312 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482418⊢ rpow x (q + r) = rpow x q * rpow x r
  rw [rpow_eq_lim_ratPow hx hq', rpow_eq_lim_ratPow hx hr', rpow_eq_lim_ratPow hx hq'r', ←(lim_mul h1 h2).2, mul_coe]x:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n + r' n)).TendsTo (q + r)h1:?_mvar.485289 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482409h2:?_mvar.485312 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482418⊢ (lim ↑fun n => x ^ ↑(q' n + r' n)) = lim ↑fun n => x ^ ↑(q' n) * x ^ ↑(r' n)
  rcongr ne_a.e_a.hx:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n + r' n)).TendsTo (q + r)h1:?_mvar.485289 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482409h2:?_mvar.485312 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482418n:ℕ⊢ x ^ ↑(q' n + r' n) = x ^ ↑(q' n) * x ^ ↑(r' n); rw [←rpow_add]e_a.e_a.hx:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n + r' n)).TendsTo (q + r)h1:?_mvar.485289 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482409h2:?_mvar.485312 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482418n:ℕ⊢ x ^ ↑(q' n + r' n) = x ^ (↑(q' n) + ↑(r' n))e_a.e_a.h.hxx:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n + r' n)).TendsTo (q + r)h1:?_mvar.485289 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482409h2:?_mvar.485312 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482418n:ℕ⊢ 0 < x; simpe_a.e_a.h.hxx:ℝhx:x > 0q:ℝr:ℝq':ℕ → ℚhq':(↑fun n => ↑(q' n)).TendsTo qr':ℕ → ℚhr':(↑fun n => ↑(r' n)).TendsTo rhq'r':(↑fun n => ↑(q' n + r' n)).TendsTo (q + r)h1:?_mvar.485289 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482409h2:?_mvar.485312 := Chapter6.ratPow_continuous _fvar.482401 _fvar.482418n:ℕ⊢ 0 < x; linarithAll goals completed! 🐙

Твердження 6.7.3(b) / Вправа 6.7.1

theorem declaration uses 'sorry'Real.ratPow_ratPow {x:ℝ} (hx: x > 0) (q r:ℝ) : rpow (rpow x q) r = rpow x (q*r) := byx:ℝhx:x > 0q:ℝr:ℝ⊢ rpow (rpow x q) r = rpow x (q * r)
  sorryAll goals completed! 🐙

Твердження 6.7.3(c) / Вправа 6.7.1

theorem declaration uses 'sorry'Real.ratPow_neg {x:ℝ} (hx: x > 0) (q:ℝ) : rpow x (-q) = 1 / rpow x q := byx:ℝhx:x > 0q:ℝ⊢ rpow x (-q) = 1 / rpow x q
  sorryAll goals completed! 🐙

Твердження 6.7.3(d) / Вправа 6.7.1

theorem declaration uses 'sorry'Real.ratPow_mono {x y:ℝ} (hx: x > 0) (hy: y > 0) {q:ℝ} (h: q > 0) : x > y ↔ rpow x q > rpow y q := byx:ℝy:ℝhx:x > 0hy:y > 0q:ℝh:q > 0⊢ x > y ↔ rpow x q > rpow y q
  sorryAll goals completed! 🐙

Твердження 6.7.3(e) / Вправа 6.7.1

theorem declaration uses 'sorry'Real.ratPow_mono_of_gt_one {x:ℝ} (hx: x > 1) {q r:ℝ} : rpow x q > rpow x r ↔ q > r := byx:ℝhx:x > 1q:ℝr:ℝ⊢ rpow x q > rpow x r ↔ q > r
  sorryAll goals completed! 🐙

Твердження 6.7.3(e) / Вправа 6.7.1

theorem declaration uses 'sorry'Real.ratPow_mono_of_lt_one {x:ℝ} (hx0: 0 < x) (hx: x < 1) {q r:ℝ} : rpow x q > rpow x r ↔ q < r := byx:ℝhx0:0 < xhx:x < 1q:ℝr:ℝ⊢ rpow x q > rpow x r ↔ q < r
  sorryAll goals completed! 🐙

Твердження 6.7.3(f) / Вправа 6.7.1

theorem declaration uses 'sorry'Real.ratPow_mul {x y:ℝ} (hx: x > 0) (hy: y > 0) (q:ℝ) : rpow (x*y) q = rpow x q * rpow y q := byx:ℝy:ℝhx:x > 0hy:y > 0q:ℝ⊢ rpow (x * y) q = rpow x q * rpow y q
  sorryAll goals completed! 🐙

end Chapter6