theorem extracted_formal_statement_69.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)]
  (C : ℝ) (left : C > 0) (hC : ∀ (n : ℕ), ‖e₁ n‖ ≤ C * ‖(OptProblem.A₁ E₂) (e₁ n)‖)
  (h_original_goal : Tendsto e₁ atTop (𝓝 0)) : Tendsto (fun n => C * ‖(OptProblem.A₁ E₂) (e₁ n)‖) atTop (𝓝 0) := sorry

State before:
case intro.intro
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
C : ℝ
left✝ : C > 0
hC : ∀ (n : ℕ), ‖e₁ n‖ ≤ C * ‖(OptProblem.A₁ E₂) (e₁ n)‖
⊢ Tendsto e₁ atTop (𝓝 0)

Tactic: apply squeeze_zero_norm
    intro n
    exact hC n

State after:
[case intro.intro.a
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
 C : ℝ
 left✝ : C > 0
 hC : ∀ (n : ℕ), ‖e₁ n‖ ≤ C * ‖(OptProblem.A₁ E₂) (e₁ n)‖
 ⊢ Tendsto (fun n => C * ‖(OptProblem.A₁ E₂) (e₁ n)‖) atTop (𝓝 0)]

Relevant lemmas:
squeeze_zero_norm : ∀ {α : Type u_2} {E : Type u_5} [inst : SeminormedAddGroup E] {f : α → E} {a : α → ℝ}
  {t₀ : Filter α}, (∀ (n : α), ‖f n‖ ≤ a n) → Tendsto a t₀ (𝓝 0) → Tendsto f t₀ (𝓝 0)