theorem extracted_formal_statement_2.{u_2, u_1} {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E]
  [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F]
  [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] {f : E → ℝ} {g : F → ℝ} {H : WithLp 2 (E × F) → ℝ}
  {x0 : E} {y0 : F} {l : NNReal} {alg : BCD f g H l x0 y0} (γ : ℝ) (hγ : γ > 1) (ck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l))
  (dk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)) (lbdψ : BddBelow (ψ '' univ)) (A : ℝ)
  (hA : Tendsto (fun n => ∑ k ∈ Finset.range n, ‖z (k + 1) - z k‖ ^ 2) atTop (𝓝 A))
  (h_original_goal : Tendsto (fun k => ‖z (k + 1) - z k‖) atTop (𝓝 0)) :
  ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖z (n + 1) - z n‖ < ε := sorry

State before:
case intro
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
f : E → ℝ
g : F → ℝ
H : WithLp 2 (E × F) → ℝ
x0 : E
y0 : F
l : NNReal
alg : BCD f g H l x0 y0
γ : ℝ
hγ : γ > 1
ck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)
dk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)
lbdψ : BddBelow (ψ '' univ)
A : ℝ
hA : Tendsto (fun n => ∑ k ∈ Finset.range n, ‖z (k + 1) - z k‖ ^ 2) atTop (𝓝 A)
⊢ Tendsto (fun k => ‖z (k + 1) - z k‖) atTop (𝓝 0)

Tactic: rw [Metric.tendsto_atTop] at hA
    simp [dist_eq_norm] at hA
    rw [Metric.tendsto_atTop]
    simp [dist_zero_right]
    have SqConver : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖alg.z (n + 1) - alg.z n‖ ^ 2 < ε :=
  by
  intro ε εge0
  specialize hA (ε / 2)
  rcases(hA (by linarith [εge0])) with ⟨N, hNεhalf⟩
  use N; intro n ngeN
  have eq' :
    ‖alg.z (n + 1) - alg.z n‖ ^ 2 =
      (∑ k ∈ Finset.range (n + 1), ‖alg.z (k + 1) - alg.z k‖ ^ 2 - A) -
        (∑ k ∈ Finset.range n, ‖alg.z (k + 1) - alg.z k‖ ^ 2 - A) :=
    by rw [sub_sub_sub_comm]; simp; rw [Finset.sum_range_succ_sub_sum]
  obtain hNεhalf' := hNεhalf (n + 1) (by linarith [ngeN])
  have hNεhalf1 : ∑ k ∈ Finset.range (n + 1), ‖alg.z (k + 1) - alg.z k‖ ^ 2 - A < ε / 2 := by rw [abs_lt] at hNεhalf';
    exact hNεhalf'.right
  have hNεhalf2 : ∑ k ∈ Finset.range n, ‖alg.z (k + 1) - alg.z k‖ ^ 2 - A > -(ε / 2) :=
    by
    specialize hNεhalf n ngeN
    rw [abs_lt] at hNεhalf; exact hNεhalf.left
  rw [eq']; linarith [hNεhalf1, hNεhalf1]

State after:
[case intro
 E : Type u_1
 F : Type u_2
 inst✝⁵ : NormedAddCommGroup E
 inst✝⁴ : InnerProductSpace ℝ E
 inst✝³ : CompleteSpace E
 inst✝² : NormedAddCommGroup F
 inst✝¹ : InnerProductSpace ℝ F
 inst✝ : CompleteSpace F
 f : E → ℝ
 g : F → ℝ
 H : WithLp 2 (E × F) → ℝ
 x0 : E
 y0 : F
 l : NNReal
 alg : BCD f g H l x0 y0
 γ : ℝ
 hγ : γ > 1
 ck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)
 dk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)
 lbdψ : BddBelow (ψ '' univ)
 A : ℝ
 hA : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → |∑ k ∈ Finset.range n, ‖z (k + 1) - z k‖ ^ 2 - A| < ε
 SqConver : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖z (n + 1) - z n‖ ^ 2 < ε
 ⊢ ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖z (n + 1) - z n‖ < ε]

Relevant lemmas:
error