theorem extracted_formal_statement_298.{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 :
    (fun n => ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n))) = fun n =>
      -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))) :
  Tendsto (fun n => -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))) atTop
    (𝓝 (-ADMM.ρ E₁ E₂ F • 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_kkt
this :
  (fun n => ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n))) = fun n =>
    -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))
⊢ Tendsto (fun n => -ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n) - ADMM.x₂ E₁ F (↑(φ₁ n) - 1))) atTop
    (𝓝 (-ADMM.ρ E₁ E₂ F • 0))

Tactic: apply Filter.Tendsto.const_smul (sub_Seq_converge_zero₃' fullrank₁ fullrank₂)

State after:

Relevant lemmas:
Filter.Tendsto.const_smul : ∀ {M : Type u_1} {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace α]
  [inst_10 : SMul M α] [inst_11 : ContinuousConstSMul M α] {f : β → α} {l : Filter β} {a : α},
  Tendsto f l (𝓝 a) → ∀ (c : M), Tendsto (fun x => c • f x) l (𝓝 (c • a))