theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
    (hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
    ∃ i : ℕ,
      ∀ j ≥ i,
        abv
            (((∑ k in range j, a k) * ∑ k in range j, b k) -
              ∑ n in range j, ∑ m in range (n + 1), a m * b (n - m)) <
          ε :=
  let ⟨Q, hQ⟩ := CauSeq.bounded ⟨_, hb⟩
  let ⟨P, hP⟩ := CauSeq.bounded ⟨_, ha⟩
  have hP0 : 0 < P := lt_of_le_of_lt (abs_nonneg _) (hP 0)
  have hPε0 : 0 < ε / (2 * P) := div_pos ε0 (mul_pos (show (2 : α) > 0 by norm_num) hP0)
  let ⟨N, hN⟩ := CauSeq.cauchy₂ ⟨_, hb⟩ hPε0
  have hQε0 : 0 < ε / (4 * Q) :=
    div_pos ε0 (mul_pos (show (0 : α) < 4 by norm_num) (lt_of_le_of_lt (abv_nonneg _ _) (hQ 0)))
  let ⟨M, hM⟩ := CauSeq.cauchy₂ ⟨_, ha⟩ hQε0
  ⟨2 * (max N M + 1), fun K hK => by
    have h₁ :
      (∑ m in range K, ∑ k in range (m + 1), a k * b (m - k)) =
        ∑ m in range K, ∑ n in range (K - m), a m * b n :=
      by simpa using sum_range_diag_flip K fun m n => a m * b n
    have h₂ :
      (fun i => ∑ k in range (K - i), a i * b k) = fun i => a i * ∑ k in range (K - i), b k := by
      simp [Finset.mul_sum]
    have h₃ :
      (∑ i in range K, a i * ∑ k in range (K - i), b k) =
        (∑ i in range K, a i * ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) +
          ∑ i in range K, a i * ∑ k in range K, b k :=
      by rw [← sum_add_distrib]; simp [(mul_add _ _ _).symm]
    have two_mul_two : (4 : α) = 2 * 2 := by norm_num
    have hQ0 : Q ≠ 0 := fun h => by simp [h, lt_irrefl] at hQε0
    have h2Q0 : 2 * Q ≠ 0 := mul_ne_zero two_ne_zero hQ0
    have hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε := by
      rw [← div_div, div_mul_cancel _ (Ne.symm (ne_of_lt hP0)), two_mul_two, mul_assoc, ← div_div,
        div_mul_cancel _ h2Q0, add_halves]
    have hNMK : max N M + 1 < K :=
      lt_of_lt_of_le (by rw [two_mul]; exact lt_add_of_pos_left _ (Nat.succ_pos _)) hK
    have hKN : N < K :=
      calc
        N ≤ max N M := le_max_left _ _
        _ < max N M + 1 := (Nat.lt_succ_self _)
        _ < K := hNMK

    have hsumlesum :
      (∑ i in range (max N M + 1),
          abv (a i) * abv ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) ≤
        ∑ i in range (max N M + 1), abv (a i) * (ε / (2 * P))
    · gcongr with m hmJ
      exact le_of_lt
            (hN (K - m)
              (le_tsub_of_add_le_left
                (le_trans
                  (by
                    rw [two_mul]
                    gcongr
                    · exact le_of_lt (mem_range.1 hmJ)
                    · exact le_trans (le_max_left _ _) (le_of_lt (lt_add_one _)))
                  hK))
              K (le_of_lt hKN))
    have hsumltP : (∑ n in range (max N M + 1), abv (a n)) < P :=
      calc
        (∑ n in range (max N M + 1), abv (a n)) = |∑ n in range (max N M + 1), abv (a n)| :=
          Eq.symm (abs_of_nonneg (sum_nonneg fun x _ => abv_nonneg abv (a x)))
        _ < P := hP (max N M + 1)

    rw [h₁, h₂, h₃, sum_mul, ← sub_sub, sub_right_comm, sub_self, zero_sub, abv_neg abv]
    refine' lt_of_le_of_lt (abv_sum_le_sum_abv _ _) _
    suffices
      (∑ i in range (max N M + 1),
            abv (a i) * abv ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) +
          ((∑ i in range K, abv (a i) * abv ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) -
            ∑ i in range (max N M + 1),
              abv (a i) * abv ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) <
        ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) by
      rw [hε] at this
      simpa [abv_mul abv] using this
    gcongr
    · 

Proof state:
case h₁
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property := hb } i) < Q
P : α
hP : ∀ (i : ℕ), abs' (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } i) < P
hP0 : 0 < P
hPε0 : 0 < ε / (2 * P)
N : ℕ
hN :
  ∀ j ≥ N,
    ∀ k ≥ N,
      abv
          (↑{ val := fun m => ∑ n in range m, b n, property := hb } j -
            ↑{ val := fun m => ∑ n in range m, b n, property := hb } k) <
        ε / (2 * P)
hQε0 : 0 < ε / (4 * Q)
M : ℕ
hM :
  ∀ j ≥ M,
    ∀ k ≥ M,
      abs'
          (↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } j -
            ↑{ val := fun m => ∑ n in range m, abv (a n), property := ha } k) <
        ε / (4 * Q)
K : ℕ
hK : K ≥ 2 * (max N M + 1)
h₁ : ∑ m in range K, ∑ k in range (m + 1), a k * b (m - k) = ∑ m in range K, ∑ n in range (K - m), a m * b n
h₂ : (fun i => ∑ k in range (K - i), a i * b k) = fun i => a i * ∑ k in range (K - i), b k
h₃ :
  ∑ i in range K, a i * ∑ k in range (K - i), b k =
    ∑ i in range K, a i * (∑ k in range (K - i), b k - ∑ k in range K, b k) + ∑ i in range K, a i * ∑ k in range K, b 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 in range (max N M + 1), abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k) ≤
    ∑ i in range (max N M + 1), abv (a i) * (ε / (2 * P))
hsumltP : ∑ n in range (max N M + 1), abv (a n) < P
⊢ ∑ i in range (max N M + 1), abv (a i) * abv (∑ k in range (K - i), b k - ∑ k in range K, b k) < ε / (2 * P) * P

Next tactic:
exact lt_of_le_of_lt hsumlesum
          (by rw [← sum_mul, mul_comm]; gcongr)