-- Module: Mathlib.Algebra.Order.CauSeq.BigOperators
Goal:
α : Type u_1
inst : Field α
inst_1 : LinearOrder α
inst_2 : IsStrictOrderedRing α
inst_3 : Archimedean α
f : ℕ → α
a : α
m : ℕ
ham : ∀ n ≥ m, |f n| ≤ a
hnm : ∀ n ≥ m, f n.succ ≤ f n
ε : α
ε0 : ε > 0
let let₁ let₂ : ∀ (k : ℕ) (hk : a ≤ k • ε), ∃ (i : ℕ), ∀ j ≥ i, |f j - f i| < ε
let₃ let₄ : ∀ (k : ℕ) (hk : a ≤ k • ε) (h : ∃ (l : ℕ), ∀ n ≥ m, a - l • ε < f n), ∃ (i : ℕ), ∀ j ≥ i, |f j - f i| < ε
let₅ :
  ∀ (k : ℕ) (hk : a ≤ k • ε) (h : ∃ (l : ℕ), ∀ n ≥ m, a - l • ε < f n),
    let l := Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h;
    ∀ (hl : ∀ n ≥ m, f n > a - l • ε), ∃ (i : ℕ), ∀ j ≥ i, |f j - f i| < ε
let₆ :
  ∀ (k : ℕ) (hk : a ≤ k • ε) (h : ∃ (l : ℕ), ∀ n ≥ m, a - l • ε < f n),
    let l := Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h;
    ∀ (hl : ∀ n ≥ m, f n > a - l • ε) (hl0 : l ≠ 0), ∃ (i : ℕ), ∀ j ≥ i, |f j - f i| < ε
let₇ :
  ∀ (k : ℕ) (hk : a ≤ k • ε) (h : ∃ (l : ℕ), ∀ n ≥ m, a - l • ε < f n),
    let l := Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h;
    ∀ (hl : ∀ n ≥ m, f n > a - l • ε) (hl0 : l ≠ 0) (i : ℕ)
      (hi : ¬(i ≥ m → a - (Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h).pred • ε < f i)),
      ∃ (i : ℕ), ∀ j ≥ i, |f j - f i| < ε
let₈ :
  ∀ (k : ℕ) (hk : a ≤ k • ε) (h : ∃ (l : ℕ), ∀ n ≥ m, a - l • ε < f n),
    let l := Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h;
    ∀ (hl : ∀ n ≥ m, f n > a - l • ε) (hl0 : l ≠ 0) (i : ℕ)
      (hi : i ≥ m ∧ f i ≤ a - (Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h).pred • ε),
      ∃ (i : ℕ), ∀ j ≥ i, |f j - f i| < ε
let₉ let₁₀ :
  ∀ (k : ℕ) (hk : a ≤ k • ε) (h : ∃ (l : ℕ), ∀ n ≥ m, a - l • ε < f n),
    let l := Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h;
    ∀ (hl : ∀ n ≥ m, f n > a - l • ε) (hl0 : l ≠ 0) (i : ℕ)
      (hi : i ≥ m ∧ f i ≤ a - (Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h).pred • ε) (j : ℕ)
      (hj : j ≥ i), |f j - f i| < ε
let₁₁ :
  ∀ (k : ℕ) (hk : a ≤ k • ε) (h : ∃ (l : ℕ), ∀ n ≥ m, a - l • ε < f n),
    let l := Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h;
    ∀ (hl : ∀ n ≥ m, f n > a - l • ε) (hl0 : l ≠ 0) (i : ℕ)
      (hi : i ≥ m ∧ f i ≤ a - (Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h).pred • ε) (j : ℕ)
      (hj : j ≥ i) (hfij : f j ≤ f i), |f j - f i| < ε
let₁₂ :
  ∀ (k : ℕ) (hk : a ≤ k • ε) (h : ∃ (l : ℕ), ∀ n ≥ m, a - l • ε < f n),
    let l := Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h;
    ∀ (hl : ∀ n ≥ m, f n > a - l • ε) (hl0 : l ≠ 0) (i : ℕ)
      (hi : i ≥ m ∧ f i ≤ a - (Nat.find (p := fun (n : ℕ) ↦ ∀ n_1 ≥ m, a - n • ε < f n_1) h).pred • ε) (j : ℕ)
      (hj : j ≥ i) (hfij : f j ≤ f i), f i < f j + ε
⊢ ∃ (i : ℕ), ∀ j ≥ i, |f j - f i| < ε

Tactic:
classical
  let ⟨k, hk⟩ := Archimedean.arch a ε0
  have h : ∃ l, ∀ n ≥ m, a - l • ε < f n :=
    ⟨k + k + 1, fun n hnm ↦
      lt_of_lt_of_le (show a - (k + (k + 1)) • ε < -|f n| from
          lt_neg.1 <| (ham n hnm).trans_lt
              (by
                rw [neg_sub, lt_sub_iff_add_lt, add_nsmul, add_nsmul, one_nsmul]
                exact add_lt_add_of_le_of_lt hk (lt_of_le_of_lt hk (lt_add_of_pos_right _ ε0))))
        (neg_le.2 <| abs_neg (f n) ▸ le_abs_self _)⟩
  let l := Nat.find h
  have hl : ∀ n : ℕ, n ≥ m → f n > a - l • ε := Nat.find_spec h
  have hl0 : l ≠ 0 := fun hl0 ↦
    not_lt_of_ge (ham m le_rfl)
      (lt_of_lt_of_le (by have := hl m (le_refl m); simpa [hl0] using this) (le_abs_self (f m)))
  obtain ⟨i, hi⟩ := not_forall.1 (Nat.find_min h (Nat.pred_lt hl0))
  rw [Classical.not_imp, not_lt] at hi
  exists i
  intro j hj
  have hfij : f j ≤ f i := (Nat.rel_of_forall_rel_succ_of_le_of_le (· ≥ ·) hnm hi.1 hj).le
  rw [abs_of_nonpos (sub_nonpos.2 hfij), neg_sub, sub_lt_iff_lt_add']
  calc
    f i ≤ a - Nat.pred l • ε := hi.2
    _ = a - l • ε + ε := by
      conv =>
        rhs
        rw [← Nat.succ_pred_eq_of_pos (Nat.pos_of_ne_zero hl0), succ_nsmul, sub_add,
          add_sub_cancel_right]
    _ < f j + ε := by gcongr; exact hl _ <| hi.1.trans hj

Premises used: neg_sub, sub_lt_iff_lt_add', not_lt, Classical.not_imp