theorem extracted_formal_statement_7.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {f : E → ℝ} {f' : E → E} {xm x₀ : E} {alg : Gradient_Descent_fix_stepsize f f' x₀}
  (hfun : ConvexOn ℝ univ f)
  (step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)) :
  ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀ := sorry

State before:
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
f : E → ℝ
f' : E → E
xm x₀ : E
alg : Gradient_Descent_fix_stepsize f f' x₀
hfun : ConvexOn ℝ univ f
step₂ : Gradient_Descent_fix_stepsize.a f f' x₀ ≤ 1 / ↑(Gradient_Descent_fix_stepsize.l f f' x₀)
k : ℕ
⊢ ↑(Gradient_Descent_fix_stepsize.l f f' x₀) ≤ 1 / Gradient_Descent_fix_stepsize.a f f' x₀

Tactic: rw [le_one_div alg.step₁] at step₂
    exact step₂
    linarith [alg.hl]

State after:

Relevant lemmas:
le_one_div : ∀ {α : Type u_2} [inst : LinearOrderedSemifield α] {a b : α}, 0 < a → 0 < b → (a ≤ 1 / b ↔ b ≤ 1 / a)