theorem extracted_formal_statement_50.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  {f : E → ℝ} (xm x₀ : E) {alg : subgradient_method f x₀} (hm : IsMinOn f Set.univ xm)
  (ha' : Tendsto (subgradient_method.a f x₀) atTop (𝓝 0))
  (ha'' : Tendsto (fun k => (Finset.range (k + 1)).sum (subgradient_method.a f x₀)) atTop atTop)
  (h₁ :
    Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) atTop (𝓝 0))
  (h₁' :
    Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm + f xm) atTop
      (𝓝 (0 + f xm))) :
  Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}) atTop
    (𝓝 (f xm)) := sorry

State before:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
f : E → ℝ
xm x₀ : E
alg : subgradient_method f x₀
hm : IsMinOn f Set.univ xm
ha' : Tendsto (subgradient_method.a f x₀) atTop (𝓝 0)
ha'' : Tendsto (fun k => (Finset.range (k + 1)).sum (subgradient_method.a f x₀)) atTop atTop
h₁ : Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm) atTop (𝓝 0)
h₁' :
  Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x} - f xm + f xm) atTop
    (𝓝 (0 + f xm))
⊢ Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}) atTop (𝓝 (f xm))

Tactic: simp at h₁'
    simp
    apply h₁'

State after:

Relevant lemmas: