theorem extracted_formal_statement_248.{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} [inst_9 : Setting E₁ E₂ F admm admm_kkt]
  (eq :
    (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖) = fun n =>
      ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - OptProblem.b E₁ E₂‖)
  (h_original_goal :
    Tendsto (fun n => ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - OptProblem.b E₁ E₂‖)
      atTop (𝓝 0)) :
  Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖) atTop (𝓝 0) := sorry

State before:
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
inst✝ : Setting E₁ E₂ F admm admm_kkt
eq :
  (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖) = fun n =>
    ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - OptProblem.b E₁ E₂‖
⊢ Tendsto (fun n => ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - OptProblem.b E₁ E₂‖)
    atTop (𝓝 0)

Tactic: rw [← eq]

State after:
[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
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 eq :
   (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖) = fun n =>
     ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - OptProblem.b E₁ E₂‖
 ⊢ Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖) atTop (𝓝 0)]

Relevant lemmas: