theorem extracted_formal_statement_315.{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]
  (this_1 :
    (fun n => v ↑(φ₁ n)) = fun n =>
      -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
          (ADMM.y E₁ E₂ ↑(φ₁ n) +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
              ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n)))))
  (h :
    Tendsto
      (fun n =>
        ADMM.y E₁ E₂ ↑(φ₁ n) +
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))))
      atTop (𝓝 y''))
  (this :
    Tendsto (⇑(-ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))) (𝓝 y'')
      (𝓝 (-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) y'')))
  (h_original_goal :
    Tendsto
      (fun n =>
        -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
            (ADMM.y E₁ E₂ ↑(φ₁ n) +
              ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
                ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n)))))
      atTop (𝓝 (-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) y''))) :
  Tendsto
    (fun n =>
      ADMM.y E₁ E₂ ↑(φ₁ n) +
        ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))))
    atTop (𝓝 y'') := 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
this✝ :
  (fun n => v ↑(φ₁ n)) = fun n =>
    -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
        (ADMM.y E₁ E₂ ↑(φ₁ n) +
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))))
h :
  Tendsto
    (fun n =>
      ADMM.y E₁ E₂ ↑(φ₁ n) +
        ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))))
    atTop (𝓝 y'')
this :
  Tendsto (⇑(-ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))) (𝓝 y'')
    (𝓝 (-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) y''))
⊢ Tendsto
    (fun n =>
      -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
          (ADMM.y E₁ E₂ ↑(φ₁ n) +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
              ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n)))))
    atTop (𝓝 (-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) y''))

Tactic: apply Filter.tendsto_iff_seq_tendsto.1 this

State after:
[case 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_kkt
 this✝ :
   (fun n => v ↑(φ₁ n)) = fun n =>
     -(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
         (ADMM.y E₁ E₂ ↑(φ₁ n) +
           ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
             ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))))
 h :
   Tendsto
     (fun n =>
       ADMM.y E₁ E₂ ↑(φ₁ n) +
         ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))))
     atTop (𝓝 y'')
 this :
   Tendsto (⇑(-ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))) (𝓝 y'')
     (𝓝 (-(ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) y''))
 ⊢ Tendsto
     (fun n =>
       ADMM.y E₁ E₂ ↑(φ₁ n) +
         ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))))
     atTop (𝓝 y'')]

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