theorem extracted_formal_statement_5.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ) :
  ConvexOn ℝ univ h := sorry

State before:
case hf
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
⊢ ConvexOn ℝ univ h

Tactic: apply alg.hconv

State after:

Relevant lemmas: