-- Mathlib/Algebra/Order/CauSeq/BigOperators.lean
theorem _root_.cauchy_product (ha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n))
    (hb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n) (ε : α) (ε0 : 0 < ε) :
    ∃ i : ℕ, ∀ j ≥ i,
      abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k -
        ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε

Proof state:
case h.h
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (N ⊔ M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : N ⊔ M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (N ⊔ M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (N ⊔ M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (N ⊔ M + 1), abv (f n) < P
i✝ : ℕ
a✝ : i✝ ∈ filter (fun i => N ⊔ M + 1 ≤ i) (range K)
⊢ abv (∑ k ∈ range (K - i✝), g k - ∑ k ∈ range K, g k) ≤ 2 * Q

Tactic:
rw [sub_eq_add_neg]

Result:
case h.h
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (N ⊔ M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : N ⊔ M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (N ⊔ M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (N ⊔ M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (N ⊔ M + 1), abv (f n) < P
i✝ : ℕ
a✝ : i✝ ∈ filter (fun i => N ⊔ M + 1 ≤ i) (range K)
⊢ abv (∑ k ∈ range (K - i✝), g k + -∑ k ∈ range K, g k) ≤ 2 * Q

Proof state:
case h₂
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (N ⊔ M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : N ⊔ M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (N ⊔ M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (N ⊔ M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (N ⊔ M + 1), abv (f n) < P
⊢ ∑ k ∈ filter (fun k => N ⊔ M + 1 ≤ k) (range K), abv (f k) * abv (∑ k ∈ range (K - k), g k - ∑ k ∈ range K, g k) <
    ε / (4 * Q) * (2 * Q)

Tactic:
calc
  (∑ i ∈ range K with max N M + 1 ≤ i,
        abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) ≤
      ∑ i ∈ range K with max N M + 1 ≤ i, abv (f i) * (2 * Q) := by
      gcongr
      rw [sub_eq_add_neg]
      refine le_trans (abv_add _ _ _) ?_
      rw [two_mul, abv_neg abv]
      gcongr <;> exact le_of_lt (hQ _)
  _ < ε / (4 * Q) * (2 * Q) := by
      rw [← sum_mul, ← sum_range_sub_sum_range (le_of_lt hNMK)]
      have := lt_of_le_of_lt (abv_nonneg _ _) (hQ 0)
      gcongr
      exact (le_abs_self _).trans_lt <|
        hM _ ((Nat.le_succ_of_le (le_max_right _ _)).trans hNMK.le) _ <|
          Nat.le_succ_of_le <| le_max_right _ _

Result:
no goals

Proof state:
case h.h
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f g : ℕ → β
ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)
hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n
ε : α
ε0 : 0 < ε
P : α
hP : ∀ (i : ℕ), |∑ n ∈ range i, abv (f n)| < P
Q : α
hQ : ∀ (i : ℕ), abv (∑ n ∈ range i, g n) < Q
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN : ∀ j ≥ N, ∀ k ≥ N, abv (∑ n ∈ range j, g n - ∑ n ∈ range k, g n) < ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM : ∀ j ≥ M, ∀ k ≥ M, |∑ n ∈ range j, abv (f n) - ∑ n ∈ range k, abv (f n)| < ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (N ⊔ M + 1)
h₁ : ∑ m ∈ range K, ∑ k ∈ range (m + 1), f k * g (m - k) = ∑ m ∈ range K, ∑ n ∈ range (K - m), f m * g n
h₂ : (fun i => ∑ k ∈ range (K - i), f i * g k) = fun i => f i * ∑ k ∈ range (K - i), g k
h₃ :
  ∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
    ∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) + ∑ i ∈ range K, f i * ∑ k ∈ range K, g k
two_mul_two : 4 = 2 * 2
hQ0 : Q ≠ 0
h2Q0 : 2 * Q ≠ 0
hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε
hNMK : N ⊔ M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (N ⊔ M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (N ⊔ M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (N ⊔ M + 1), abv (f n) < P
i✝ : ℕ
a✝ : i✝ ∈ filter (fun i => N ⊔ M + 1 ≤ i) (range K)
⊢ abv (∑ k ∈ range (K - i✝), g k) + abv (∑ k ∈ range K, g k) ≤ Q + Q

Tactic:
gcongr <;> exact le_of_lt (hQ _)

Result:
no goals