theorem extracted_formal_statement_32.{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}
  (h : Tendsto (fun n => ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)‖ ^ 2) atTop (𝓝 0))
  (this : Tendsto (fun n => √(‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)‖ ^ 2)) atTop (𝓝 √0))
  (h_original_goal : Tendsto (fun n => ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)‖) atTop (𝓝 0)) :
  Tendsto (fun n => ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F 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
h : Tendsto (fun n => ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)‖ ^ 2) atTop (𝓝 0)
this : Tendsto (fun n => √(‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)‖ ^ 2)) atTop (𝓝 √0)
⊢ Tendsto (fun n => ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)‖) atTop (𝓝 0)

Tactic: rw [Real.sqrt_zero] at this
    simp [Real.sqrt_sq] at this
    simp

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
 h : Tendsto (fun n => ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n)‖ ^ 2) atTop (𝓝 0)
 this : Tendsto (fun n => ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n)‖) atTop (𝓝 0)
 ⊢ Tendsto (fun n => ‖(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n)‖) atTop (𝓝 0)]

Relevant lemmas:
Real.sqrt_zero : √0 = 0
Real.sqrt_sq : ∀ {x : ℝ}, 0 ≤ x → √(x ^ 2) = x