Аналіз I, Глава 7.4

I have attempted to make the translation as faithful a paraphrasing as possible of the original text. When there is a choice between a more idiomatic Lean solution and a more faithful translation, I have generally chosen the latter. In particular, there will be places where the Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided doing so.

Main constructions and results of this section:

Rearrangement of non-negative or absolutely convergent series

namespace Chapter7

theorem Series.sum_eq_sum (b:ℕ → ℝ) {N:ℤ} (hN: N ≥ 0) : ∑ n ∈ Finset.Icc 0 N, (if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := byb:ℕ → ℝN:ℤhN:N ≥ 0⊢ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n
      convert Finset.sum_image (g := fun n:ℕ ↦ (n:ℤ)) _h.e'_2.hb:ℕ → ℝN:ℤhN:N ≥ 0⊢ Finset.Icc 0 N = Finset.image (fun n => ↑n) (Finset.Iic N.toNat)convert_5b:ℕ → ℝN:ℤhN:N ≥ 0⊢ ∀ x ∈ Finset.Iic N.toNat, ∀ y ∈ Finset.Iic N.toNat, (fun n => ↑n) x = (fun n => ↑n) y → x = y
      .h.e'_2.hb:ℕ → ℝN:ℤhN:N ≥ 0⊢ Finset.Icc 0 N = Finset.image (fun n => ↑n) (Finset.Iic N.toNat) ext xh.e'_2.h.hb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ x ∈ Finset.Icc 0 N ↔ x ∈ Finset.image (fun n => ↑n) (Finset.Iic N.toNat)
        simph.e'_2.h.hb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N ↔ ∃ a ≤ N.toNat, ↑a = x
        constructorh.e'_2.h.h.mpb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N → ∃ a ≤ N.toNat, ↑a = xh.e'_2.h.h.mprb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ (∃ a ≤ N.toNat, ↑a = x) → 0 ≤ x ∧ x ≤ N
        .h.e'_2.h.h.mpb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N → ∃ a ≤ N.toNat, ↑a = x intro ⟨ hpos, hx ⟩h.e'_2.h.h.mpb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤhpos:0 ≤ xhx:x ≤ N⊢ ∃ a ≤ N.toNat, ↑a = x
          use x.toNathb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤhpos:0 ≤ xhx:x ≤ N⊢ x.toNat ≤ N.toNat ∧ ↑x.toNat = x
          omegaAll goals completed! 🐙
        intro ⟨ a, ⟨ ha, hb ⟩ ⟩h.e'_2.h.h.mprb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤa:ℕha:a ≤ N.toNathb:↑a = x⊢ 0 ≤ x ∧ x ≤ N
        simp [←hb]h.e'_2.h.h.mprb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤa:ℕha:a ≤ N.toNathb:↑a = x⊢ ↑a ≤ N
        omegaAll goals completed! 🐙
      simpAll goals completed! 🐙

Твердження 7.4.1

theorem Series.converges_of_permute_nonneg {a:ℕ → ℝ} (ha: (a:Series).nonneg) (hconv: (a:Series).converges)
  {f: ℕ → ℕ} (hf: Function.Bijective f) :
    (fun n ↦ a (f n) : Series).converges ∧ (a:Series).sum = (fun n ↦ a (f n) : Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  -- цей доказ написан так, щоб співпадати із структурою орігінального тексту.
  set af : ℕ → ℝ := fun n ↦ a (f n)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  have haf : (af:Series).nonneg := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    unfold nonneg at ha ⊢a:ℕ → ℝha:∀ (n : ℤ), { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq n ≥ 0hconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)⊢ ∀ (n : ℤ), { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0
    intro na:ℕ → ℝha:∀ (n : ℤ), { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq n ≥ 0hconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)n:ℤ⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0; by_cases h : n ≥ 0posa:ℕ → ℝha:∀ (n : ℤ), { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq n ≥ 0hconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)n:ℤh:n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0nega:ℕ → ℝha:∀ (n : ℤ), { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq n ≥ 0hconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)n:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0
    all_goals simp [af, h]All goals completed! 🐙
    specialize ha (f n.toNat)posa:ℕ → ℝhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)n:ℤh:n ≥ 0ha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑(f n.toNat) ≥ 0⊢ 0 ≤ a (f n.toNat)
    aesopAll goals completed! 🐙
  set S := (a:Series).partiala:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  set T := (af:Series).partiala:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  have hSmono : Monotone S := Series.partial_of_nonneg haa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone S⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  have hTmono : Monotone T := Series.partial_of_nonneg hafa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone T⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  set L := iSup Sa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup S⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  set L' := iSup Ta:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup T⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  have hSBound : ∃ Q, ∀ N, S N ≤ Q := (converges_of_nonneg_iff ha).mp hconva:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Q⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  suffices: (∃ Q, ∀ M, T M ≤ Q) ∧ L = L'a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sumthisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Q⊢ (∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'
  .a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum have Ssum : L = (a:Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      apply (sum_of_converges _).symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesTo L
      simp [convergesTo, L]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ Filter.Tendsto { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial Filter.atTop
  (nhds (iSup S))
      apply tendsto_atTop_isLUB  hSmono _a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ IsLUB (Set.range S) (iSup S)
      apply isLUB_csSupnea:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ (Set.range S).NonemptyHa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ BddAbove (Set.range S)
      .nea:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ (Set.range S).Nonempty use (S 0)ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'⊢ S 0 ∈ Set.range S; aesopAll goals completed! 🐙
      obtain ⟨ Q, hQ ⟩ := hSBoundH.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup Tthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Q:ℝhQ:∀ (N : ℤ), S N ≤ Q⊢ BddAbove (Set.range S)
      use Qha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup Tthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Q:ℝhQ:∀ (N : ℤ), S N ≤ Q⊢ Q ∈ upperBounds (Set.range S)
      simp [upperBounds, hQ]All goals completed! 🐙
    have Tsum : L' = (af:Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      apply (sum_of_converges _).symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'
      simp [convergesTo, L']a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ Filter.Tendsto { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial Filter.atTop
  (nhds (iSup T))
      apply tendsto_atTop_isLUB  hTmono _a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ IsLUB (Set.range T) (iSup T)
      apply isLUB_csSupnea:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ (Set.range T).NonemptyHa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ BddAbove (Set.range T)
      .nea:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ (Set.range T).Nonempty use (T 0)ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ T 0 ∈ Set.range T; aesopAll goals completed! 🐙
      obtain ⟨ Q, hQ ⟩ := this.1H.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumQ:ℝhQ:∀ (M : ℤ), T M ≤ Q⊢ BddAbove (Set.range T)
      use Qha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumQ:ℝhQ:∀ (M : ℤ), T M ≤ Q⊢ Q ∈ upperBounds (Set.range T)
      simp [upperBounds, hQ]All goals completed! 🐙
    rw [←Ssum, ←Tsum]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumTsum:L' = { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.converges ∧ L = L'
    simp [this.2]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumTsum:L' = { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum⊢ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.converges
    rw [converges_of_nonneg_iff haf]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ Qthis:(∃ Q, ∀ (M : ℤ), T M ≤ Q) ∧ L = L'Ssum:L = { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumTsum:L' = { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum⊢ ∃ M, ∀ (N : ℤ), { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial N ≤ M
    convert this.1All goals completed! 🐙
  have hTL (M:ℤ) : T M ≤ L := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    by_cases hM : M ≥ 0posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0⊢ T M ≤ Lnega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:¬M ≥ 0⊢ T M ≤ L
    swapnega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:¬M ≥ 0⊢ T M ≤ Lposa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0⊢ T M ≤ L
    .nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:¬M ≥ 0⊢ T M ≤ L have hM' : M < 0 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum linarithAll goals completed! 🐙
      simp [T, Series.partial, hM']nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:¬M ≥ 0hM':M < 0⊢ 0 ≤ L
      convert le_ciSup (f := S) ?_ (-1)nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:¬M ≥ 0hM':M < 0⊢ BddAbove (Set.range S)
      simp [BddAbove, Set.Nonempty, upperBounds, hSBound]All goals completed! 🐙
    set Y := Finset.Iic M.toNatposa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNat⊢ T M ≤ L
    have hN : ∃ N, ∀ m ∈ Y, f m ≤ N := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      use (Y.image f).sup idha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNat⊢ ∀ m ∈ Y, f m ≤ (Finset.image f Y).sup id
      intro m hmha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatm:ℕhm:m ∈ Y⊢ f m ≤ (Finset.image f Y).sup id
      apply Finset.le_sup (f := id)ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatm:ℕhm:m ∈ Y⊢ f m ∈ Finset.image f Y
      simpha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatm:ℕhm:m ∈ Y⊢ ∃ a ∈ Y, f a = f m; use mAll goals completed! 🐙
    obtain ⟨ N, hN ⟩ := hNpos.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ T M ≤ L
    calc
      _ = ∑ m ∈ Y, af m := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ T M = ∑ m ∈ Y, af m
        simp [T, Series.partial, af]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (∑ n ∈ Finset.Icc 0 M, if 0 ≤ n then a (f n.toNat) else 0) = ∑ n ∈ Y, a (f n)
        exact sum_eq_sum af hMAll goals completed! 🐙
      _ = ∑ n ∈ f '' Y, a n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ m ∈ Y, af m = ∑ n ∈ (f '' ↑Y).toFinset, a n
        symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ n ∈ (f '' ↑Y).toFinset, a n = ∑ m ∈ Y, af m
        convert Finset.sum_image _h.e'_2.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (f '' ↑Y).toFinset = Finset.image f Yconvert_6a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ DecidableEq ℕconvert_9a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∀ x ∈ Y, ∀ y ∈ Y, f x = f y → x = y
        .h.e'_2.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (f '' ↑Y).toFinset = Finset.image f Y simpAll goals completed! 🐙
        .convert_6a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ DecidableEq ℕ infer_instanceAll goals completed! 🐙
        intro x hx y hy hxyconvert_9a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ Nx:ℕhx:x ∈ Yy:ℕhy:y ∈ Yhxy:f x = f y⊢ x = y
        exact hf.injective hxyAll goals completed! 🐙
      _ ≤ ∑ n ∈ Finset.Iic N, a n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ n ∈ (f '' ↑Y).toFinset, a n ≤ ∑ n ∈ Finset.Iic N, a n
        apply Finset.sum_le_sum_of_subset_of_nonnegha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (f '' ↑Y).toFinset ⊆ Finset.Iic Nhfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∀ i ∈ Finset.Iic N, i ∉ (f '' ↑Y).toFinset → 0 ≤ a i
        ·ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (f '' ↑Y).toFinset ⊆ Finset.Iic N intro n hnha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ Nn:ℕhn:n ∈ (f '' ↑Y).toFinset⊢ n ∈ Finset.Iic N
          simp at hn ⊢ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ Nn:ℕhn:∃ a ∈ Y, f a = n⊢ n ≤ N
          obtain ⟨ a, ha, rfl ⟩ := hnh.intro.introa✝:ℕ → ℝha✝:{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a✝ (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ Na:ℕha:a ∈ Y⊢ f a ≤ N
          exact hN a haAll goals completed! 🐙
        intro i _ _hfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ Ni:ℕa✝¹:i ∈ Finset.Iic Na✝:i ∉ (f '' ↑Y).toFinset⊢ 0 ≤ a i
        specialize ha ihfa:ℕ → ℝhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ Ni:ℕa✝¹:i ∈ Finset.Iic Na✝:i ∉ (f '' ↑Y).toFinsetha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑i ≥ 0⊢ 0 ≤ a i
        simp at hahfa:ℕ → ℝhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ Ni:ℕa✝¹:i ∈ Finset.Iic Na✝:i ∉ (f '' ↑Y).toFinsetha:0 ≤ a i⊢ 0 ≤ a i; exact haAll goals completed! 🐙
      _ = S N := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ n ∈ Finset.Iic N, a n = S ↑N
        simp [S, Series.partial]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ∑ n ∈ Finset.Iic N, a n = ∑ n ∈ Finset.Icc 0 ↑N, if 0 ≤ n then a n.toNat else 0
        symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ (∑ n ∈ Finset.Icc 0 ↑N, if 0 ≤ n then a n.toNat else 0) = ∑ n ∈ Finset.Iic N, a n
        exact sum_eq_sum (N:=N) a (bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ ↑N ≥ 0 positivityAll goals completed! 🐙)
      _ ≤ L := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ S ↑N ≤ L
        apply le_ciSup _ (N:ℤ)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QM:ℤhM:M ≥ 0Y:Finset ℕ := Finset.Iic M.toNatN:ℕhN:∀ m ∈ Y, f m ≤ N⊢ BddAbove (Set.range S)
        simp [BddAbove, Set.Nonempty, upperBounds, hSBound]All goals completed! 🐙
  have hTbound : ∃ Q, ∀ M, T M ≤ Q := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum use LAll goals completed! 🐙
  simp [hTbound]thisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ Q⊢ L = L'
  have hL'L : L' ≤ L := ciSup_le hTLthisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ L⊢ L = L'
  have hSL' (N:ℤ) : S N ≤ L' := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    by_cases hN : N ≥ 0posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0⊢ S N ≤ L'nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:¬N ≥ 0⊢ S N ≤ L'
    swapnega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:¬N ≥ 0⊢ S N ≤ L'posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0⊢ S N ≤ L'
    .nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:¬N ≥ 0⊢ S N ≤ L' have hN' : N < 0 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum linarithAll goals completed! 🐙
      simp [S, Series.partial, hN']nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:¬N ≥ 0hN':N < 0⊢ 0 ≤ L'
      convert le_ciSup (f := T) ?_ (-1)nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:¬N ≥ 0hN':N < 0⊢ BddAbove (Set.range T)
      simp [BddAbove, Set.Nonempty, upperBounds, hTbound]All goals completed! 🐙
    set X := Finset.Iic N.toNatposa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNat⊢ S N ≤ L'
    have hM : ∃ M, ∀ n ∈ X, ∃ m, f m = n ∧ m ≤ M := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      use (X.preimage f (Set.injOn_of_injective hf.1)).sup idha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNat⊢ ∀ n ∈ X, ∃ m, f m = n ∧ m ≤ (X.preimage f ⋯).sup id
      intro n hnha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatn:ℕhn:n ∈ X⊢ ∃ m, f m = n ∧ m ≤ (X.preimage f ⋯).sup id
      obtain ⟨ m, hm ⟩ := hf.2 nh.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatn:ℕhn:n ∈ Xm:ℕhm:f m = n⊢ ∃ m, f m = n ∧ m ≤ (X.preimage f ⋯).sup id
      refine ⟨ m, hm, ?_ ⟩h.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatn:ℕhn:n ∈ Xm:ℕhm:f m = n⊢ m ≤ (X.preimage f ⋯).sup id
      apply Finset.le_sup (f := id)h.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatn:ℕhn:n ∈ Xm:ℕhm:f m = n⊢ m ∈ X.preimage f ⋯
      simp [Finset.mem_preimage, hm, hn]All goals completed! 🐙
    obtain ⟨ M, hM ⟩ := hMpos.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ M⊢ S N ≤ L'
    have sum_eq_sum (b:ℕ → ℝ) {N:ℤ} (hN: N ≥ 0) : ∑ n ∈ Finset.Icc 0 N, (if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.converges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
      convert Finset.sum_image (g := fun n:ℕ ↦ (n:ℤ)) _h.e'_2.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0⊢ Finset.Icc 0 N = Finset.image (fun n => ↑n) (Finset.Iic N.toNat)convert_5a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0⊢ ∀ x ∈ Finset.Iic N.toNat, ∀ y ∈ Finset.Iic N.toNat, (fun n => ↑n) x = (fun n => ↑n) y → x = y
      .h.e'_2.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0⊢ Finset.Icc 0 N = Finset.image (fun n => ↑n) (Finset.Iic N.toNat) ext xh.e'_2.h.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ x ∈ Finset.Icc 0 N ↔ x ∈ Finset.image (fun n => ↑n) (Finset.Iic N.toNat); simp [X]h.e'_2.h.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N ↔ ∃ a ≤ N.toNat, ↑a = x
        constructorh.e'_2.h.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N → ∃ a ≤ N.toNat, ↑a = xh.e'_2.h.h.mpra:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ (∃ a ≤ N.toNat, ↑a = x) → 0 ≤ x ∧ x ≤ N
        .h.e'_2.h.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ 0 ≤ x ∧ x ≤ N → ∃ a ≤ N.toNat, ↑a = x intro ⟨ hpos, hx ⟩h.e'_2.h.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤhpos:0 ≤ xhx:x ≤ N⊢ ∃ a ≤ N.toNat, ↑a = x
          use x.toNatha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤhpos:0 ≤ xhx:x ≤ N⊢ x.toNat ≤ N.toNat ∧ ↑x.toNat = x; omegaAll goals completed! 🐙
        intro ⟨ a, ⟨ ha, hb ⟩ ⟩h.e'_2.h.h.mpra✝:ℕ → ℝha✝:{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a✝ (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤa:ℕha:a ≤ N.toNathb:↑a = x⊢ 0 ≤ x ∧ x ≤ N
        simp [←hb]h.e'_2.h.h.mpra✝:ℕ → ℝha✝:{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a✝ (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N✝.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤa:ℕha:a ≤ N.toNathb:↑a = x⊢ ↑a ≤ N; omegaAll goals completed! 🐙
      simpAll goals completed! 🐙
    calc
      _ = ∑ n ∈ X, a n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ S N = ∑ n ∈ X, a n
        simp [S, Series.partial]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then a n.toNat else 0) = ∑ n ∈ X, a n
        exact sum_eq_sum a hNAll goals completed! 🐙
      _ = ∑ n ∈ Finset.image f ((Finset.Iic M).filter (fun m ↦ f m ∈  X)), a n := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ ∑ n ∈ X, a n = ∑ n ∈ Finset.image f ({m ∈ Finset.Iic M | f m ∈ X}), a n
        congre_sa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ X = Finset.image f ({m ∈ Finset.Iic M | f m ∈ X}); ext ne_s.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nn:ℕ⊢ n ∈ X ↔ n ∈ Finset.image f ({m ∈ Finset.Iic M | f m ∈ X}); simpe_s.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nn:ℕ⊢ n ∈ X ↔ ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = n
        constructore_s.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nn:ℕ⊢ n ∈ X → ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = ne_s.h.mpra:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nn:ℕ⊢ (∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = n) → n ∈ X
        .e_s.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nn:ℕ⊢ n ∈ X → ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = n intro he_s.h.mpa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nn:ℕh:n ∈ X⊢ ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = n
          obtain ⟨ m, hm, hm' ⟩ := hM n he_s.h.mp.intro.introa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nn:ℕh:n ∈ Xm:ℕhm:f m = nhm':m ≤ M⊢ ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = n
          use mha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nn:ℕh:n ∈ Xm:ℕhm:f m = nhm':m ≤ M⊢ (m ≤ M ∧ f m ∈ X) ∧ f m = n; simp [hm', hm, h]All goals completed! 🐙
        intro ⟨ m, ⟨ hm, hfmX⟩ , hfm ⟩e_s.h.mpra:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nn:ℕm:ℕhm:m ≤ MhfmX:f m ∈ Xhfm:f m = n⊢ n ∈ X
        simp [←hfm, hfmX]All goals completed! 🐙
      _ ≤ ∑ m ∈ Finset.Iic M, af m := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ ∑ n ∈ Finset.image f ({m ∈ Finset.Iic M | f m ∈ X}), a n ≤ ∑ m ∈ Finset.Iic M, af m
        rw [Finset.sum_image _]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ ∑ x ∈ Finset.Iic M with f x ∈ X, a (f x) ≤ ∑ m ∈ Finset.Iic M, af ma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ ∀ x ∈ {m ∈ Finset.Iic M | f m ∈ X}, ∀ y ∈ {m ∈ Finset.Iic M | f m ∈ X}, f x = f y → x = y
        .a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ ∑ x ∈ Finset.Iic M with f x ∈ X, a (f x) ≤ ∑ m ∈ Finset.Iic M, af m apply Finset.sum_le_sum_of_subset_of_nonnegha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ {m ∈ Finset.Iic M | f m ∈ X} ⊆ Finset.Iic Mhfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ ∀ i ∈ Finset.Iic M, i ∉ {m ∈ Finset.Iic M | f m ∈ X} → 0 ≤ a (f i)
          .ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ {m ∈ Finset.Iic M | f m ∈ X} ⊆ Finset.Iic M intro mha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nm:ℕ⊢ m ∈ {m ∈ Finset.Iic M | f m ∈ X} → m ∈ Finset.Iic M; simpha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nm:ℕ⊢ m ≤ M → f m ∈ X → m ≤ M; tautoAll goals completed! 🐙
          intro i _ _hfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b ni:ℕa✝¹:i ∈ Finset.Iic Ma✝:i ∉ {m ∈ Finset.Iic M | f m ∈ X}⊢ 0 ≤ a (f i)
          specialize haf ihfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b ni:ℕa✝¹:i ∈ Finset.Iic Ma✝:i ∉ {m ∈ Finset.Iic M | f m ∈ X}haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq ↑i ≥ 0⊢ 0 ≤ a (f i)
          simp at hafhfa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)S:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b ni:ℕa✝¹:i ∈ Finset.Iic Ma✝:i ∉ {m ∈ Finset.Iic M | f m ∈ X}haf:0 ≤ af i⊢ 0 ≤ a (f i)
          exact hafAll goals completed! 🐙
        intro x _ y _ hxya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b nx:ℕa✝¹:x ∈ {m ∈ Finset.Iic M | f m ∈ X}y:ℕa✝:y ∈ {m ∈ Finset.Iic M | f m ∈ X}hxy:f x = f y⊢ x = y
        exact hf.injective hxyAll goals completed! 🐙
      _ = T M := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ ∑ m ∈ Finset.Iic M, af m = T ↑M
        simp [T, Series.partial, af]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ ∑ n ∈ Finset.Iic M, a (f n) = ∑ n ∈ Finset.Icc 0 ↑M, if 0 ≤ n then a (f n.toNat) else 0
        symma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ (∑ n ∈ Finset.Icc 0 ↑M, if 0 ≤ n then a (f n.toNat) else 0) = ∑ n ∈ Finset.Iic M, a (f n)
        apply sum_eq_sum af (bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ ↑M ≥ 0 positivityAll goals completed! 🐙)
      _ ≤ L' := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ T ↑M ≤ L'
        apply le_ciSup _ (M:ℤ)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Msum_eq_sum:∀ (b : ℕ → ℝ) {N : ℤ}, N ≥ 0 → (∑ n ∈ Finset.Icc 0 N, if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ Finset.Iic N.toNat, b n⊢ BddAbove (Set.range T)
        simp [BddAbove, Set.Nonempty, upperBounds, hTbound]All goals completed! 🐙
  have hLL' : L ≤ L' := ciSup_le hSL'thisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.nonneghconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesf:ℕ → ℕhf:Function.Bijective faf:ℕ → ℝ := fun n => a (f n)haf:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.nonnegS:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partialT:ℤ → ℝ := { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partialhSmono:Monotone ShTmono:Monotone TL:ℝ := iSup SL':ℝ := iSup ThSBound:∃ Q, ∀ (N : ℤ), S N ≤ QhTL:∀ (M : ℤ), T M ≤ LhTbound:∃ Q, ∀ (M : ℤ), T M ≤ QhL'L:L' ≤ LhSL':∀ (N : ℤ), S N ≤ L'hLL':L ≤ L'⊢ L = L'
  linarithAll goals completed! 🐙

Приклад 7.4.2

theorem declaration uses 'sorry'Series.zeta_2_converges : (fun n:ℕ ↦ 1/(n+1:ℝ)^2 : Series).converges := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => 1 / (↑n + 1) ^ 2) n.toNat else 0, vanish := ⋯ }.converges sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.permuted_zeta_2_converges :
  (fun n:ℕ ↦ if Even n then 1/(n+2:ℝ)^2 else 1/(n:ℝ)^2 : Series).converges := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => if Even n then 1 / (↑n + 2) ^ 2 else 1 / ↑n ^ 2) n.toNat else 0,
    vanish := ⋯ }.converges
    sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.permuted_zeta_2_eq_zeta_2 :
  (fun n:ℕ ↦ if Even n then 1/(n+2:ℝ)^2 else 1/(n:ℝ)^2 : Series).sum = (fun n:ℕ ↦ 1/(n+1:ℝ)^2 : Series).sum := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => if Even n then 1 / (↑n + 2) ^ 2 else 1 / ↑n ^ 2) n.toNat else 0,
      vanish := ⋯ }.sum =
  { m := 0, seq := fun n => if n ≥ 0 then (fun n => 1 / (↑n + 1) ^ 2) n.toNat else 0, vanish := ⋯ }.sum
    sorryAll goals completed! 🐙

Твердження 7.4.3 (Rearrangement of series)

theorem declaration uses 'sorry'Series.absConverges_of_permute {a:ℕ → ℝ} (ha : (a:Series).absConverges)
  {f: ℕ → ℕ} (hf: Function.Bijective f) :
    (fun n ↦ a (f n):Series).absConverges  ∧ (a:Series).sum = (fun n ↦ a (f n) : Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  -- цей доказ написан так, щоб співпадати із структурою орігінального тексту.
  set L := (a:Series).abs.suma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sum⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  have hconv := converges_of_absConverges haa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.converges⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  unfold absConverges at haa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.converges⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  have habs : (fun n ↦ |a (f n)| : Series).converges ∧ L = (fun n ↦ |a (f n)| : Series).sum := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    convert converges_of_permute_nonneg (a := fun n ↦ |a n|) _ _ hf using 3h.e'_2.h.e'_2.h.e'_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.converges⊢ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs =
  { m := 0, seq := fun n => if n ≥ 0 then |a n.toNat| else 0, vanish := ⋯ }convert_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.converges⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.nonnegconvert_2a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.converges⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.converges
    .h.e'_2.h.e'_2.h.e'_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.converges⊢ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs =
  { m := 0, seq := fun n => if n ≥ 0 then |a n.toNat| else 0, vanish := ⋯ } simph.e'_2.h.e'_2.h.e'_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.converges⊢ (fun n => if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = fun n => if 0 ≤ n then |a n.toNat| else 0; ext nh.e'_2.h.e'_2.h.e'_1.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤ⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0
      by_cases h: n ≥ 0posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:n ≥ 0⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:¬n ≥ 0⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0 <;>posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:n ≥ 0⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:¬n ≥ 0⊢ (if 0 ≤ n then |if 0 ≤ n then a n.toNat else 0| else 0) = if 0 ≤ n then |a n.toNat| else 0 simp [h]All goals completed! 🐙
    .convert_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.converges⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.nonneg intro nconvert_1a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤ⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0
      by_cases h: n ≥ 0posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0 <;>posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a n|) n.toNat else 0, vanish := ⋯ }.seq n ≥ 0 simp [h]All goals completed! 🐙
    convert ha with nh.e'_1.h.e'_2.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤ⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0
    by_cases h: n ≥ 0posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:n ≥ 0⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:¬n ≥ 0⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0 <;>posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:n ≥ 0⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesn:ℤh:¬n ≥ 0⊢ (if n ≥ 0 then (fun n => |a n|) n.toNat else 0) =
  if h : n ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0 simp [h]All goals completed! 🐙
  set L' := (a:Series).suma:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  L' = { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
  set af : ℕ → ℝ := fun n ↦ a (f n)a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.absConverges ∧
  L' = { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum
  suffices: (af:Series).convergesTo L'a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.absConverges ∧
  L' = { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sumthisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'
  .a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.absConverges ∧
  L' = { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.sum simp [sum_of_converges this, absConverges]a:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'⊢ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.abs.converges
    convert habs.1 with nh.e'_1.h.e'_2.ha:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤ⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0
    by_cases h: n ≥ 0posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤh:n ≥ 0⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤh:¬n ≥ 0⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0 <;>posa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤh:n ≥ 0⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0nega:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)this:{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.convergesTo L'n:ℤh:¬n ≥ 0⊢ (if h : n ≥ { m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.m then
    (fun n => |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.seq ↑n|) ⟨n, h⟩
  else 0) =
  if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0 simp [h, af]All goals completed! 🐙
  simp [convergesTo, LinearOrderedAddCommGroup.tendsto_nhds]thisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)⊢ ∀ (ε : ℝ),
  0 < ε →
    ∃ a,
      ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  intro ε hεthisa:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.convergesf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < ε⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  rw [converges_iff_tail_decay] at hathisa:ℕ → ℝha:∀ ε > 0,
  ∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
    ∀ p ≥ N,
      ∀ q ≥ N,
        |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ εf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < ε⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  specialize ha (ε/2) (half_pos hε)thisa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εha:∃ N ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.m,
  ∀ p ≥ N,
    ∀ q ≥ N,
      |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  obtain ⟨ N₁, hN₁, ha ⟩ := hathis.intro.introa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤhN₁:N₁ ≥ { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.mha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  simp at hN₁this.intro.introa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  have : ∃ N ≥ N₁, |(a:Series).partial N - L'| < ε/2 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    replace hconv := convergesTo_sum hconva:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁hconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergesTo
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum⊢ ∃ N ≥ N₁, |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2
    simp [convergesTo, LinearOrderedAddCommGroup.tendsto_nhds] at hconva:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁hconv:∀ (ε : ℝ),
  0 < ε →
    ∃ a_2,
      ∀ (b : ℤ),
        a_2 ≤ b →
          |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
                { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
            ε⊢ ∃ N ≥ N₁, |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2
    specialize hconv (ε/2) (half_pos hε)a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁hconv:∃ a_1,
  ∀ (b : ℤ),
    a_1 ≤ b →
      |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
            { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
        ε / 2⊢ ∃ N ≥ N₁, |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2
    obtain ⟨ N, hN ⟩ := hconvintroa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:∀ (b : ℤ),
  N ≤ b →
    |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
          { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
      ε / 2⊢ ∃ N ≥ N₁, |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2
    use max N N₁, le_max_right _ _righta:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:∀ (b : ℤ),
  N ≤ b →
    |{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial b -
          { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
      ε / 2⊢ |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial (max N N₁) - L'| < ε / 2
    specialize hN (max N N₁) (le_max_left _ _)righta:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:|{ m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.partial (max N N₁) -
      { m := 0, seq := fun n => if 0 ≤ n then a n.toNat else 0, vanish := ⋯ }.sum| <
  ε / 2⊢ |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial (max N N₁) - L'| < ε / 2
    convert hNAll goals completed! 🐙
  obtain ⟨ N, hN, hN2 ⟩ := thisthis.intro.intro.intro.introa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  have hNpos : N ≥ 0 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum linarithAll goals completed! 🐙
  have finv : ℕ → ℕ := Function.invFun fthis.intro.intro.intro.introa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕ⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  have : ∃ M, ∀ n ≤ N.toNat, finv n ≤ M := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    use ((Finset.Iic (N.toNat)).image finv).sup idha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕ⊢ ∀ n ≤ N.toNat, finv n ≤ (Finset.image finv (Finset.Iic N.toNat)).sup id
    intro n hnha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕn:ℕhn:n ≤ N.toNat⊢ finv n ≤ (Finset.image finv (Finset.Iic N.toNat)).sup id
    apply Finset.le_sup (f := id)ha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕn:ℕhn:n ≤ N.toNat⊢ finv n ∈ Finset.image finv (Finset.Iic N.toNat)
    simp [Finset.mem_image]ha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕn:ℕhn:n ≤ N.toNat⊢ ∃ a ≤ N.toNat, finv a = finv n
    use nAll goals completed! 🐙
  obtain ⟨ M, hM ⟩ := thisthis.intro.intro.intro.intro.introa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ M⊢ ∃ a, ∀ (b : ℤ), a ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  use Mha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ M⊢ ∀ (b : ℤ), ↑M ≤ b → |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial b - L'| < ε
  intro M' hM'ha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'⊢ |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial M' - L'| < ε
  have hM'_pos : M' ≥ 0 := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum linarithAll goals completed! 🐙
  have why : Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNat := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum
    sorryAll goals completed! 🐙
  set X : Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial M' - L'| < ε
  have claim : ∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a n := calc
    _ = ∑ n ∈ Finset.image f (Finset.Iic M'.toNat), a n := bya:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ ∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.image f (Finset.Iic M'.toNat), a n
      symma:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ ∑ n ∈ Finset.image f (Finset.Iic M'.toNat), a n = ∑ m ∈ Finset.Iic M'.toNat, a (f m)
      apply Finset.sum_imageaa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ ∀ x ∈ Finset.Iic M'.toNat, ∀ y ∈ Finset.Iic M'.toNat, f x = f y → x = y
      intro x _ y _ hxyaa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatx:ℕa✝¹:x ∈ Finset.Iic M'.toNaty:ℕa✝:y ∈ Finset.Iic M'.toNathxy:f x = f y⊢ x = y
      exact hf.1 hxyAll goals completed! 🐙
    _ = _ := bya:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ ∑ n ∈ Finset.image f (Finset.Iic M'.toNat), a n = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a n
      convert Finset.sum_union _ using 2h.e'_2.ha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ Finset.image f (Finset.Iic M'.toNat) = Finset.Iic N.toNat ∪ Xconvert_7a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ DecidableEq ℕconvert_8a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ Disjoint (Finset.Iic N.toNat) X
      .h.e'_2.ha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ Finset.image f (Finset.Iic M'.toNat) = Finset.Iic N.toNat ∪ X simp [X, why]All goals completed! 🐙
      .convert_7a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ DecidableEq ℕ infer_instanceAll goals completed! 🐙
      rw [Finset.disjoint_right]convert_8a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNat⊢ ∀ ⦃a : ℕ⦄, a ∈ X → a ∉ Finset.Iic N.toNat
      intro n hnconvert_8a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatn:ℕhn:n ∈ X⊢ n ∉ Finset.Iic N.toNat
      simp only [X, Finset.mem_sdiff] at hnconvert_8a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatn:ℕhn:n ∈ Finset.image f (Finset.Iic M'.toNat) ∧ n ∉ Finset.Iic N.toNat⊢ n ∉ Finset.Iic N.toNat
      tautoAll goals completed! 🐙
  obtain ⟨ q', hq ⟩ := X.bddAboveh.introa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑X⊢ |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial M' - L'| < ε
  set q := max q' N.toNath.introa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNat⊢ |{ m := 0, seq := fun n => if 0 ≤ n then af n.toNat else 0, vanish := ⋯ }.partial M' - L'| < ε
  have why2 : X ⊆ Finset.Icc (N.toNat+1) q := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:Function.Bijective f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges ∧
  { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sum =
    { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.sum sorryAll goals completed! 🐙
  have claim2 : |∑ n ∈ X, a n| ≤ ε/2 := calc
    _ ≤ ∑ n ∈ X, |a n| := Finset.abs_sum_le_sum_abs a X
    _ ≤ ∑ n ∈ Finset.Icc (N.toNat+1) q, |a n| := bya:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ ∑ n ∈ X, |a n| ≤ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n|
      apply Finset.sum_le_sum_of_subset_of_nonneg why2a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ ∀ i ∈ Finset.Icc (N.toNat + 1) q, i ∉ X → 0 ≤ |a i|
      simpAll goals completed! 🐙
    _ ≤ ε/2 := bya:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n| ≤ ε / 2
      convert ha (N.toNat+1) (bya:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ ↑N.toNat + 1 ≥ N₁ omegaAll goals completed! 🐙) q (bya:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ ↑q ≥ N₁ omegaAll goals completed! 🐙)
      simp [hNpos]h.e'_3a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n| =
  |∑ x ∈ Finset.Icc (N + 1) ↑q, if 0 ≤ x then |if 0 ≤ x then a x.toNat else 0| else 0|
      rw [abs_of_nonneg (by positivity)]h.e'_3a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n| =
  ∑ x ∈ Finset.Icc (N + 1) ↑q, if 0 ≤ x then |if 0 ≤ x then a x.toNat else 0| else 0
      symmh.e'_3a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ (∑ x ∈ Finset.Icc (N + 1) ↑q, if 0 ≤ x then |if 0 ≤ x then a x.toNat else 0| else 0) =
  ∑ n ∈ Finset.Icc (N.toNat + 1) q, |a n|
      convert Finset.sum_image (g := fun (n:ℕ) ↦ (n:ℤ)) _ using 2h.e'_2.ha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ Finset.Icc (N + 1) ↑q = Finset.image (fun n => ↑n) (Finset.Icc (N.toNat + 1) q)h.e'_3.convert_5a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ ∀ x ∈ Finset.Icc (N.toNat + 1) q, ∀ y ∈ Finset.Icc (N.toNat + 1) q, (fun n => ↑n) x = (fun n => ↑n) y → x = y
      .h.e'_2.ha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) q⊢ Finset.Icc (N + 1) ↑q = Finset.image (fun n => ↑n) (Finset.Icc (N.toNat + 1) q) ext xh.e'_2.h.ha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qx:ℤ⊢ x ∈ Finset.Icc (N + 1) ↑q ↔ x ∈ Finset.image (fun n => ↑n) (Finset.Icc (N.toNat + 1) q); simph.e'_2.h.ha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qx:ℤ⊢ N + 1 ≤ x ∧ x ≤ ↑q ↔ ∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = x
        constructorh.e'_2.h.h.mpa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qx:ℤ⊢ N + 1 ≤ x ∧ x ≤ ↑q → ∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = xh.e'_2.h.h.mpra:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qx:ℤ⊢ (∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = x) → N + 1 ≤ x ∧ x ≤ ↑q
        .h.e'_2.h.h.mpa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qx:ℤ⊢ N + 1 ≤ x ∧ x ≤ ↑q → ∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = x intro ⟨ hpos, hx ⟩h.e'_2.h.h.mpa:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qx:ℤhpos:N + 1 ≤ xhx:x ≤ ↑q⊢ ∃ a, (N.toNat + 1 ≤ a ∧ a ≤ q) ∧ ↑a = x
          use x.toNatha:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qx:ℤhpos:N + 1 ≤ xhx:x ≤ ↑q⊢ (N.toNat + 1 ≤ x.toNat ∧ x.toNat ≤ q) ∧ ↑x.toNat = x; omegaAll goals completed! 🐙
        intro ⟨ a, ⟨ ha, hb ⟩ ⟩h.e'_2.h.h.mpra✝:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a✝ (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a✝ (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a✝ (f n)ε:ℝhε:0 < εN₁:ℤha✝:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a✝ (f m) = ∑ n ∈ Finset.Iic N.toNat, a✝ n + ∑ n ∈ X, a✝ nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qx:ℤa:ℕha:N.toNat + 1 ≤ a ∧ a ≤ qhb:↑a = x⊢ N + 1 ≤ x ∧ x ≤ ↑q
        simp [←hb]h.e'_2.h.h.mpra✝:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a✝ (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a✝ (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a✝ (f n)ε:ℝhε:0 < εN₁:ℤha✝:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a✝ n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a✝ (f m) = ∑ n ∈ Finset.Iic N.toNat, a✝ n + ∑ n ∈ X, a✝ nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qx:ℤa:ℕha:N.toNat + 1 ≤ a ∧ a ≤ qhb:↑a = x⊢ N + 1 ≤ ↑a ∧ a ≤ q; omegaAll goals completed! 🐙
      simpAll goals completed! 🐙
  calc
    _ ≤ |(af:Series).partial M' - (a:Series).partial N| + |(a:Series).partial N - L'| := abs_sub_le _ _ _
    _ < |(af:Series).partial M' - (a:Series).partial N| + ε/2 := bya:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qclaim2:|∑ n ∈ X, a n| ≤ ε / 2⊢ |{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
        { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N| +
    |{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| <
  |{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
        { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N| +
    ε / 2
      gcongrAll goals completed! 🐙
    _ ≤ ε/2 + ε/2 := bya:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qclaim2:|∑ n ∈ X, a n| ≤ ε / 2⊢ |{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
        { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N| +
    ε / 2 ≤
  ε / 2 + ε / 2
      gcongrbca:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qclaim2:|∑ n ∈ X, a n| ≤ ε / 2⊢ |{ m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
      { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N| ≤
  ε / 2
      convert claim2h.e'_3.h.e'_4a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qclaim2:|∑ n ∈ X, a n| ≤ ε / 2⊢ { m := 0, seq := fun n => if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.partial M' -
    { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N =
  ∑ n ∈ X, a n
      simp [Series.partial, sum_eq_sum _ hM'_pos, sum_eq_sum _ hNpos]h.e'_3.h.e'_4a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qclaim2:|∑ n ∈ X, a n| ≤ ε / 2⊢ ∑ n ∈ Finset.Iic M'.toNat, af n - ∑ n ∈ Finset.Iic N.toNat, a n = ∑ n ∈ X, a n
      rw [claim]h.e'_3.h.e'_4a:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qclaim2:|∑ n ∈ X, a n| ≤ ε / 2⊢ ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a n - ∑ n ∈ Finset.Iic N.toNat, a n = ∑ n ∈ X, a n
      abelAll goals completed! 🐙
    _ = ε := bya:ℕ → ℝf:ℕ → ℕhf:Function.Bijective fL:ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.sumhconv:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.convergeshabs:{ m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.converges ∧
  L = { m := 0, seq := fun n => if n ≥ 0 then (fun n => |a (f n)|) n.toNat else 0, vanish := ⋯ }.sumL':ℝ := { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.sumaf:ℕ → ℝ := fun n => a (f n)ε:ℝhε:0 < εN₁:ℤha:∀ p ≥ N₁,
  ∀ q ≥ N₁,
    |∑ n ∈ Finset.Icc p q, { m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n| ≤ ε / 2hN₁:0 ≤ N₁N:ℤhN:N ≥ N₁hN2:|{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.partial N - L'| < ε / 2hNpos:N ≥ 0finv:ℕ → ℕM:ℕhM:∀ n ≤ N.toNat, finv n ≤ MM':ℤhM':↑M ≤ M'hM'_pos:M' ≥ 0why:Finset.image f (Finset.Iic M'.toNat) ⊇ Finset.Iic N.toNatX:Finset ℕ := Finset.image f (Finset.Iic M'.toNat) \ Finset.Iic N.toNatclaim:∑ m ∈ Finset.Iic M'.toNat, a (f m) = ∑ n ∈ Finset.Iic N.toNat, a n + ∑ n ∈ X, a nq':ℕhq:q' ∈ upperBounds ↑Xq:ℕ := max q' N.toNatwhy2:X ⊆ Finset.Icc (N.toNat + 1) qclaim2:|∑ n ∈ X, a n| ≤ ε / 2⊢ ε / 2 + ε / 2 = ε ringAll goals completed! 🐙

Приклад 7.4.4

noncomputable abbrev Series.a_7_4_4 : ℕ → ℝ := fun n ↦ (-1:ℝ)^n / (n+2)

theorem declaration uses 'sorry'Series.ex_7_4_4_conv : (a_7_4_4 : Series).converges := by⊢ { m := 0, seq := fun n => if n ≥ 0 then a_7_4_4 n.toNat else 0, vanish := ⋯ }.converges sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.ex_7_4_4_sum : (a_7_4_4 : Series).sum > 0 := by⊢ { m := 0, seq := fun n => if n ≥ 0 then a_7_4_4 n.toNat else 0, vanish := ⋯ }.sum > 0 sorryAll goals completed! 🐙

abbrev Series.f_7_4_4 : ℕ → ℕ := fun n ↦ if n % 3 = 0 then 2 * (n/3) else 2*n - (n/3) - 1

theorem declaration uses 'sorry'Series.f_7_4_4_bij : Function.Bijective f_7_4_4 := by⊢ Function.Bijective f_7_4_4 sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.ex_7_4_4'_conv : (fun n ↦ a_7_4_4 (f_7_4_4 n) :Series).converges := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a_7_4_4 (f_7_4_4 n)) n.toNat else 0, vanish := ⋯ }.converges sorryAll goals completed! 🐙

theorem declaration uses 'sorry'Series.ex_7_4_4'_sum : (fun n ↦ a_7_4_4 (f_7_4_4 n) :Series).sum < 0 := by⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a_7_4_4 (f_7_4_4 n)) n.toNat else 0, vanish := ⋯ }.sum < 0 sorryAll goals completed! 🐙

Вправа 7.4.1

theorem declaration uses 'sorry'Series.absConverges_of_subseries {a:ℕ → ℝ} (ha: (a:Series).absConverges) {f: ℕ → ℕ} (hf: StrictMono f) :
  (fun n ↦ a (f n):Series).absConverges := bya:ℕ → ℝha:{ m := 0, seq := fun n => if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.absConvergesf:ℕ → ℕhf:StrictMono f⊢ { m := 0, seq := fun n => if n ≥ 0 then (fun n => a (f n)) n.toNat else 0, vanish := ⋯ }.absConverges sorryAll goals completed! 🐙

end Chapter7