Module: Mathlib.Algebra.Order.CauSeq.BigOperators

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing 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 * (max 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 : max N M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
⊢ ∑ i ∈ range K, abv (f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k)) < ε

Tactic:
suffices
    (∑ i ∈ range (max N M + 1),
          abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) +
        ((∑ i ∈ range K, abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) -
          ∑ i ∈ range (max N M + 1),
            abv (f i) * abv ((∑ k ∈ range (K - i), g k) - ∑ k ∈ range K, g k)) <
      ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) by
    rw [hε] at this
    simpa [abv_mul abv] using this

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : Field α
inst✝³ : LinearOrder α
inst✝² : IsStrictOrderedRing 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 * (max 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 : max N M + 1 < K
hKN : N < K
hsumlesum :
  ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) ≤
    ∑ i ∈ range (max N M + 1), abv (f i) * (ε / (2 * P))
hsumltP : ∑ n ∈ range (max N M + 1), abv (f n) < P
⊢ ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) +
      (∑ i ∈ range K, abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) -
        ∑ i ∈ range (max N M + 1), abv (f i) * abv (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k)) <
    ε / (2 * P) * P + ε / (4 * Q) * (2 * Q)