theorem extracted_formal_statement_267.{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 < k + 1, f (subgradient_method.x f x₀ i) = x}) atTop (𝓝 (f xm)))
  (h_original_goal :
    Tendsto (fun k => sInf {x | ∃ i ∈ Finset.range (k + 1), f (subgradient_method.x f x₀ i) = x}) atTop (𝓝 (f xm))) :
  Tendsto (fun k => sInf {x | ∃ i < 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 < k + 1, f (subgradient_method.x f x₀ i) = x}) atTop (𝓝 (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

State after:
[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 < k + 1, f (subgradient_method.x f x₀ i) = x}) atTop (𝓝 (f xm))
 ⊢ Tendsto (fun k => sInf {x | ∃ i < k + 1, f (subgradient_method.x f x₀ i) = x}) atTop (𝓝 (f xm))]

Relevant lemmas: