theorem extracted_formal_statement_329.{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_kkt]
  (h_original_goal :
    Tendsto
      (fun n =>
        ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (φ n)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (φ n)) - OptProblem.b E₁ E₂‖)
      atTop (𝓝 ‖(OptProblem.A₁ E₂) x₁'' + (OptProblem.A₂ E₁) x₂'' - OptProblem.b E₁ E₂‖)) :
  Tendsto
    (fun x => (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (φ x)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (φ x)) - OptProblem.b E₁ E₂)
    atTop (𝓝 ((OptProblem.A₁ E₂) x₁'' + (OptProblem.A₂ E₁) x₂'' - OptProblem.b E₁ E₂)) := 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm admm_kkt
⊢ Tendsto
    (fun n => ‖(OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (φ n)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (φ n)) - OptProblem.b E₁ E₂‖)
    atTop (𝓝 ‖(OptProblem.A₁ E₂) x₁'' + (OptProblem.A₂ E₁) x₂'' - OptProblem.b E₁ E₂‖)

Tactic: apply Tendsto.norm

State after:
[case h
 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_kkt
 ⊢ Tendsto
     (fun x => (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (φ x)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (φ x)) - OptProblem.b E₁ E₂)
     atTop (𝓝 ((OptProblem.A₁ E₂) x₁'' + (OptProblem.A₂ E₁) x₂'' - OptProblem.b E₁ E₂))]

Relevant lemmas:
Filter.Tendsto.norm : ∀ {α : Type u_2} {E : Type u_5} [inst : SeminormedAddGroup E] {a : E} {l : Filter α} {f : α → E},
  Tendsto f l (𝓝 a) → Tendsto (fun x => ‖f x‖) l (𝓝 ‖a‖)