chebyshev_asymptotic''

theorem chebyshev_asymptotic'' :    ∃ (f : ℝ → ℝ),      (∀ ε > (0 : ℝ), (f =o[atTop] fun _ ↦ ε)) ∧      (∀ (x : ℝ), 2 ≤ x → IntegrableOn f (Set.Icc 2 x)) ∧      ∀ x > (0 : ℝ), θ x = x + x * (f x) := by  obtain ⟨f, hf1, inte, hf2⟩ := chebyshev_asymptotic'  refine ⟨fun t => f t / t, fun ε hε ↦ ?_, ?_, ?_⟩  · simp only [isLittleO_iff, norm_eq_abs, norm_mul, eventually_atTop,      norm_div] at hf1 ⊢    intro r hr    replace hf1 := hf1 ε hε    obtain ⟨N, hN⟩ := hf1 hr    use |N| + 1    intro x hx    have hx' : |N| + 1 ≤ |x| := by rwa [abs_of_nonneg (a := x) (le_trans (by positivity) hx)]    rw [div_le_iff₀ (lt_of_lt_of_le (by positivity) hx'), mul_assoc]    exact hN x (le_trans (le_trans (le_abs_self N) (by linarith)) hx)   · intro x hx    refine inte x hx |>.mul_continuousOn (g' := fun t : ℝ => t⁻¹)      (continuousOn_inv₀ |>.mono <| by        rintro t ⟨ht1, _⟩        simp only [Set.mem_compl_iff, Set.mem_singleton_iff]        linarith) isCompact_Icc |>.congr_fun_ae <| .of_forall <| by simp [div_eq_mul_inv]  intro x hx  rw [hf2, mul_div_cancel₀]  linarith