theorem extracted_formal_statement_57.{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)]
  (h₁ : Tendsto (⇑(OptProblem.A₁ E₂)) (𝓝 0) (𝓝 ((OptProblem.A₁ E₂) 0))) :
  Tendsto (fun n => (OptProblem.A₁ E₂) (e₁ (φ 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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm (admm_kkt₁ fullrank₁ fullrank₂)
h₁ : Tendsto (⇑(OptProblem.A₁ E₂)) (𝓝 0) (𝓝 ((OptProblem.A₁ E₂) 0))
⊢ Tendsto (fun n => (OptProblem.A₁ E₂) (e₁ (φ n))) atTop (𝓝 0)

Tactic: simp at h₁
    apply Filter.tendsto_iff_seq_tendsto.1 h₁
    apply e₁_subseq_converge_zero

State after:

Relevant lemmas:
Filter.tendsto_iff_seq_tendsto : ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {k : Filter α} {l : Filter β}
  [inst : k.IsCountablyGenerated], Tendsto f k l ↔ ∀ (x : ℕ → α), Tendsto x atTop k → Tendsto (f ∘ x) atTop l