Analysis
7.4. Rearrangement of series

Imports

import Mathlib.Tactic
import Analysis.Section_7_3

Analysis I, Section 7.4: Rearrangement of series

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 ∈ .Icc 0 N, (if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ .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 := Int.ofNat) (byb:ℕ → ℝN:ℤhN:N ≥ 0⊢ Set.InjOn Int.ofNat ↑?m.42 simpAll goals completed! 🐙)
      ext xh.e'_2.h.hb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤ⊢ x ∈ Finset.Icc 0 N ↔ x ∈ Finset.image Int.ofNat (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 ⟨ _, _ ⟩h.e'_2.h.h.mpb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤleft✝:0 ≤ xright✝:x ≤ N⊢ ∃ a ≤ N.toNat, ↑a = x; use x.toNathb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤleft✝:0 ≤ xright✝:x ≤ N⊢ x.toNat ≤ N.toNat ∧ ↑x.toNat = x; omegaAll goals completed! 🐙
      grindAll goals completed! 🐙

Proposition 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
  -- This proof is written to follow the structure of the original text.
  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
    intro na: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)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:{ 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)n:ℤh:n ≥ 0⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then af 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 := ⋯ }.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)n:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then af 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 := ⋯ }.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)n:ℤh:n ≥ 0⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then af 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 := ⋯ }.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)n:ℤh:¬n ≥ 0⊢ { m := 0, seq := fun n ↦ if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq n ≥ 0 simp [h, af]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); grindAll 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
      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 := ⋯ }.sum = L; apply sum_of_convergesha: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]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'⊢ 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 (isLUB_csSup _ _)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'⊢ (Set.range S).Nonemptya: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)
      .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'⊢ (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! 🐙
      choose Q hQ using hSBounda: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
      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 := ⋯ }.sum = L'; apply sum_of_convergesha: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']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⊢ 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 (isLUB_csSup _ _)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⊢ (Set.range T).Nonemptya: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)
      .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⊢ (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! 🐙
      choose Q hQ using this.1a: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! 🐙
    simp [←Ssum, ←Tsum, this.2, 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 0 ≤ n 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; grindAll goals completed! 🐙
    choose N hN using hNposa: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 (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⊢ Set.InjOn ?m.439 ↑?m.438 solve_by_elim [hf.injective]All goals completed! 🐙); simpAll goals completed! 🐙
      _ ≤ ∑ n ∈ .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 _ _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 ≤ Na✝¹:ℕa✝:a✝¹ ∈ (f '' ↑Y).toFinset⊢ a✝¹ ∈ Finset.Iic N; aesopAll 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✝:i ∈ Finset.Iic N⊢ i ∉ (f '' ↑Y).toFinset → 0 ≤ a 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; aesopAll 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; apply sum_eq_sum (N:=N) aa: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 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 ≤ QN: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 ≤ QN: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 ≤ QN: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 ≤ QN: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 ≤ QN: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 ≤ QN: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 ≤ QN: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 ≤ QN: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 ≤ QN: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 ≤ QN:ℤhN:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatn:ℕhn:n ∈ X⊢ ∃ m, f m = n ∧ m ≤ (X.preimage f ⋯).sup id; choose m hm using hf.2 nha: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 ≤ QN: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 ⟨ _, hm, ?_ ⟩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 ≤ QN: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)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 ≤ QN: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! 🐙
    choose M hM using hMposa: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 ≤ QN: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 ∈ .Icc 0 N, (if 0 ≤ n then b n.toNat else 0) = ∑ n ∈ .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 := Int.ofNat) (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 ≤ QN✝:ℤhN✝:N✝ ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0⊢ Set.InjOn Int.ofNat ↑?m.926 simpAll goals completed! 🐙)
      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 ≤ QN✝: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 Int.ofNat (Finset.Iic N.toNat); simph.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 ≤ QN✝: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 ≤ QN✝: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 ≤ QN✝: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 ≤ QN✝: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 ⟨ _, _ ⟩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 ≤ QN✝:ℤhN✝:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤleft✝:0 ≤ xright✝: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 ≤ QN✝:ℤhN✝:N ≥ 0X:Finset ℕ := Finset.Iic N.toNatM:ℕhM:∀ n ∈ X, ∃ m, f m = n ∧ m ≤ Mb:ℕ → ℝN:ℤhN:N ≥ 0x:ℤleft✝:0 ≤ xright✝:x ≤ N⊢ x.toNat ≤ N.toNat ∧ ↑x.toNat = x; omegaAll goals completed! 🐙
      grindAll 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 ≤ QN: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, sum_eq_sum, hN, X]All goals completed! 🐙
      _ = ∑ n ∈ ((Finset.Iic M).filter (f · ∈ X)).image f, 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 ≤ QN: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 ({x ∈ Finset.Iic M | f x ∈ 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 ≤ QN: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 ({x ∈ Finset.Iic M | f x ∈ X}); exte_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 ≤ QN: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 na✝:ℕ⊢ a✝ ∈ X ↔ a✝ ∈ Finset.image f ({x ∈ Finset.Iic M | f x ∈ 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 ≤ QN: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 na✝:ℕ⊢ a✝ ∈ X ↔ ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝; 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 ≤ QN: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 na✝:ℕ⊢ a✝ ∈ X → ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝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 ≤ QN: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 na✝:ℕ⊢ (∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝) → a✝ ∈ 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 ≤ QN: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 na✝:ℕ⊢ a✝ ∈ X → ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝ 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 ≤ QN: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 na✝:ℕh:a✝ ∈ X⊢ ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = a✝; obtain ⟨ m, rfl, hm' ⟩ := hM _ 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 ≤ QN: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:ℕhm':m ≤ Mh:f m ∈ X⊢ ∃ a, (a ≤ M ∧ f a ∈ X) ∧ f a = f m; use mAll goals completed! 🐙
        rintro ⟨ _, ⟨ _, _⟩, rfl ⟩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 ≤ QN: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 nw✝:ℕleft✝:w✝ ≤ Mright✝:f w✝ ∈ X⊢ f w✝ ∈ X; simp_allAll goals completed! 🐙
      _ ≤ ∑ m ∈ .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 ≤ QN: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 ({x ∈ Finset.Iic M | f x ∈ X}), a n ≤ ∑ m ∈ Finset.Iic M, af m
        rw [Finset.sum_image (by solve_by_elim [hf.injective])]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 ≤ QN: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 ≤ QN: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 | f x ∈ 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 ≤ QN: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 ∉ {x ∈ Finset.Iic M | f x ∈ 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 ≤ QN: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 | f x ∈ X} ⊆ Finset.Iic M aesopAll 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 ≤ QN: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✝:i ∈ Finset.Iic M⊢ i ∉ {x ∈ Finset.Iic M | f x ∈ X} → 0 ≤ a (f 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 ≤ QN: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 ∉ {x ∈ Finset.Iic M | f x ∈ 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 ≤ QN: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 ∉ {x ∈ Finset.Iic M | f x ∈ X}haf:{ m := 0, seq := fun n ↦ if n ≥ 0 then af n.toNat else 0, vanish := ⋯ }.seq ↑i ≥ 0⊢ 0 ≤ a (f i); aesopAll 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 ≤ QN: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 ≤ QN: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 ≤ QN: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 afa: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 ≤ QN: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 ≤ QN: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 ≤ QN: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! 🐙
  linarith [ciSup_le hSL', ciSup_le hTL]All goals completed! 🐙

Example 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! 🐙

Proposition 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
  -- This proof is written to follow the structure of the original text.
  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 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 grindAll 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 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 grindAll 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 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 grindAll 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
  choose N₁ hN₁ ha using ha _ (half_pos hε)thisa: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 < 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 hN₁thisa: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 < 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
    apply convergesTo_sum at hconva: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.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 → 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.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
    choose N hN using hconv _ (half_pos hε)a: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.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| <
            εN: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, (bya: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.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| <
            εN: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⊢ max N N₁ ≥ N₁ grindAll goals completed! 🐙); apply hNright.aa: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.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| <
            εN: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 ≤ max N N₁; grindAll goals completed! 🐙
  choose N hN hN2 using thisthisa: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 < 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! 🐙
  let finv : ℕ → ℕ := Function.invFun fthisa: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 < 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 := Function.invFun f⊢ ∃ 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 → 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 < 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 := Function.invFun f⊢ ∀ n ≤ N.toNat, finv n ≤ (Finset.image finv (Finset.Iic N.toNat)).sup id
    intro n hnha: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 < 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 := Function.invFun fn:ℕhn:n ≤ N.toNat⊢ finv n ≤ (Finset.image finv (Finset.Iic N.toNat)).sup id
    apply Finset.le_sup (f := id)ha: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 < 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 := Function.invFun fn:ℕhn:n ≤ N.toNat⊢ (if h : ∃ x, f x = n then h.choose else Classical.arbitrary ℕ) ∈ Finset.image finv (Finset.Iic N.toNat); simp [Finset.mem_image]ha: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 < 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 := Function.invFun fn:ℕhn:n ≤ N.toNat⊢ ∃ a ≤ N.toNat, finv a = if h : ∃ x, f x = n then h.choose else Classical.arbitrary ℕ; use n, hnrighta: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 < 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 := Function.invFun fn:ℕhn:n ≤ N.toNat⊢ finv n = if h : ∃ x, f x = n then h.choose else Classical.arbitrary ℕ; rflAll goals completed! 🐙
  choose M hM using thisthisa: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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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.Iic M'.toNat).image f ⊇ .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.Iic M'.toNat).image f \ .Iic N.toNatha: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 < 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 := Function.invFun fM: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 ∈ .Iic M'.toNat, a (f m) = ∑ n ∈ .Iic N.toNat, a n + ∑ n ∈ X, a n := calc
    _ = ∑ n ∈ (Finset.Iic M'.toNat).image f , a n := bya: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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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⊢ Set.InjOn f ↑(Finset.Iic M'.toNat); solve_by_elim [hf.1]All goals completed! 🐙
    _ = _ := bya: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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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! 🐙
  choose q' hq using X.bddAboveha: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 < 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 := Function.invFun fM: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.toNatha: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 < 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 := Function.invFun fM: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| := X.abs_sum_le_sum_abs a
    _ ≤ ∑ n ∈ .Icc (N.toNat+1) q, |a n| := bya: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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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) _ q _h.e'_3a: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 < 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 := Function.invFun fM: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| =
  |∑ n ∈ Finset.Icc (↑N.toNat + 1) ↑q, { m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n|convert_1a: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 < 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 := Function.invFun fM: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 ≥ N₁convert_2a: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 < 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 := Function.invFun fM: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 <;>h.e'_3a: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 < 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 := Function.invFun fM: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| =
  |∑ n ∈ Finset.Icc (↑N.toNat + 1) ↑q, { m := 0, seq := fun n ↦ if n ≥ 0 then a n.toNat else 0, vanish := ⋯ }.abs.seq n|convert_1a: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 < 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 := Function.invFun fM: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 ≥ N₁convert_2a: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 < 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 := Function.invFun fM: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 try omegaAll goals completed! 🐙
      simp [hNpos]h.e'_3a: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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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)) (bya: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 < 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 := Function.invFun fM: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⊢ Set.InjOn (fun n ↦ ↑n) ↑?m.1624 simpAll goals completed! 🐙) using 2
      ext xh.e'_2.h.ha: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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 < x ∧ x ≤ ↑q ↔ ∃ a, (N.toNat < a ∧ a ≤ q) ∧ ↑a = x; constructorh.e'_2.h.h.mpa: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 < 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 := Function.invFun fM: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 < x ∧ x ≤ ↑q → ∃ a, (N.toNat < a ∧ a ≤ q) ∧ ↑a = xh.e'_2.h.h.mpra: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 < 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 := Function.invFun fM: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 < a ∧ a ≤ q) ∧ ↑a = x) → N < x ∧ x ≤ ↑q
      .h.e'_2.h.h.mpa: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 < 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 := Function.invFun fM: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 < x ∧ x ≤ ↑q → ∃ a, (N.toNat < a ∧ a ≤ q) ∧ ↑a = x intro ⟨ _, _ ⟩h.e'_2.h.h.mpa: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 < 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 := Function.invFun fM: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:ℤleft✝:N < xright✝:x ≤ ↑q⊢ ∃ a, (N.toNat < a ∧ a ≤ q) ∧ ↑a = x; use x.toNatha: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 < 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 := Function.invFun fM: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:ℤleft✝:N < xright✝:x ≤ ↑q⊢ (N.toNat < x.toNat ∧ x.toNat ≤ q) ∧ ↑x.toNat = x; omegaAll goals completed! 🐙
      grindAll 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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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 → 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 < 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 := Function.invFun fM: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; grindAll goals completed! 🐙
    _ = ε := bya: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 < 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 := Function.invFun fM: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! 🐙

Example 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 4 * (n/3) + 2 * (n % 3) - 1theorem 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! 🐙

Exercise 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! 🐙

Exercise 7.4.2 : reprove Proposition 7.4.3 using Proposition 7.4.1, Proposition 7.2.14, and expressing a n as the difference of a n + |a n| and |a n|.

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 sorryAll goals completed! 🐙

end Chapter7