lemma rpow_p_mul_one_add_smoothingFn_ge :
    ∀ᶠ (n : ℕ) in atTop, ∀ i, (b i) ^ (p a b) * n ^ (p a b) * (1 + ε n)
      ≤ (r i n) ^ (p a b) * (1 + ε (r i n)) := by
  rw [Filter.eventually_all]
  intro i
  let q : ℝ → ℝ := fun x => x ^ (p a b) * (1 + ε x)
  have h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1) := by
    refine DifferentiableOn.mul
        (DifferentiableOn.mono (differentiableOn_rpow_const _) fun z hz => ?_)
        differentiableOn_one_add_smoothingFn
    rw [Set.mem_compl_singleton_iff]
    exact ne_of_gt <| zero_lt_one.trans hz
  have h_deriv_q : deriv q =O[atTop] fun x => x ^ ((p a b) - 1) := calc
    deriv q = deriv fun x => (fun z => z ^ (p a b)) x * (fun z => 1 + ε z) x := by rfl
          _ =ᶠ[atTop] fun x => deriv (fun z => z ^ (p a b)) x * (1 + ε x)
              + x ^ (p a b) * deriv (fun z => 1 + ε z) x := by
                filter_upwards [eventually_ne_atTop 0, eventually_gt_atTop 1] with x hx hx'
                rw [deriv_mul] <;> aesop
          _ =O[atTop] fun x => x ^ ((p a b) - 1) := by
                refine IsBigO.add ?left ?right
                case left => calc
                  (fun x => deriv (fun z => z ^ (p a b)) x * (1 + ε x))
                      =O[atTop] fun x => x ^ ((p a b) - 1) * (1 + ε x) := by
                        exact IsBigO.mul (isBigO_deriv_rpow_const_atTop (p a b)) (isBigO_refl _ _)
                    _ =O[atTop] fun x => x ^ ((p a b) - 1) * 1 :=
                        IsBigO.mul (isBigO_refl _ _) isEquivalent_one_add_smoothingFn_one.isBigO
                    _ = fun x => x ^ ((p a b) - 1) := by ext; rw [mul_one]
                case right => calc
                  (fun x => x ^ (p a b) * deriv (fun z => 1 + ε z) x)
                      =O[atTop] (fun x => x ^ (p a b) * x⁻¹) := by
                        exact IsBigO.mul (isBigO_refl _ _)
                          isLittleO_deriv_one_add_smoothingFn.isBigO
                    _ =ᶠ[atTop] fun x => x ^ ((p a b) - 1) := by
                        filter_upwards [eventually_gt_atTop 0] with x hx
                        rw [← Real.rpow_neg_one, ← Real.rpow_add hx, ← sub_eq_add_neg]
  have h_main_norm : (fun (n:ℕ) => ‖q (r i n) - q (b i * n)‖)
      ≤ᶠ[atTop] fun (n:ℕ) => ‖(b i) ^ (p a b) * n ^ (p a b) * (ε (b i * n) - ε n)‖ := by
    refine IsLittleO.eventuallyLE ?_
    calc
      (fun (n:ℕ) => q (r i n) - q (b i * n))
          =O[atTop] fun n => (deriv q n) * (r i n - b i * n) := by
            exact R.isBigO_apply_r_sub_b q h_diff_q
              (growsPolynomially_deriv_rpow_p_mul_one_add_smoothingFn (p a b)) i
        _ =o[atTop] fun n => (deriv q n) * (n / log n ^ 2) := by
            exact IsBigO.mul_isLittleO (isBigO_refl _ _) (R.dist_r_b i)
        _ =O[atTop] fun n => n ^ ((p a b) - 1) * (n / log n ^ 2) := by
            exact IsBigO.mul (IsBigO.nat_cast_atTop h_deriv_q) (isBigO_refl _ _)
        _ =ᶠ[atTop] fun n => n ^ (p a b) / (log n) ^ 2 := by
            filter_upwards [eventually_ne_atTop 0] with n hn
            have hn' : (n:ℝ) ≠ 0 := by positivity
            simp [← mul_div_assoc, ← Real.rpow_add_one hn']
        _ = fun (n:ℕ) => (n:ℝ) ^ (p a b) * (1 / (log n) ^ 2)   := by simp_rw [mul_div, mul_one]
        _ =Θ[atTop] fun (n:ℕ) => (b i) ^ (p a b) * n ^ (p a b) * (1 / (log n) ^ 2) := by
            refine IsTheta.symm ?_
            simp_rw [mul_assoc]
            refine IsTheta.const_mul_left ?_ (isTheta_refl _ _)
            have := R.b_pos i; positivity
        _ =Θ[atTop] fun (n:ℕ) => (b i) ^ (p a b) * n ^ (p a b) * (ε (b i * n) - ε n) := by
            exact IsTheta.symm <| IsTheta.mul (isTheta_refl _ _)
                  <| R.isTheta_smoothingFn_sub_self i
  

Proof state:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
q : ℝ → ℝ := fun x => x ^ p a b * (1 + ε x)
h_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)
h_deriv_q : deriv q =O[atTop] fun x => x ^ (p a b - 1)
h_main_norm :
  (fun n => ‖q ↑(r i n) - q (b i * ↑n)‖) ≤ᶠ[atTop] fun n => ‖b i ^ p a b * ↑n ^ p a b * (ε (b i * ↑n) - ε ↑n)‖
⊢ ∀ᶠ (x : ℕ) in atTop, b i ^ p a b * ↑x ^ p a b * (1 + ε ↑x) ≤ ↑(r i x) ^ p a b * (1 + ε ↑(r i x))

Next tactic:
have h_main : (fun (n:ℕ) => q (b i * n) - q (r i n))
      ≤ᶠ[atTop] fun (n:ℕ) => (b i) ^ (p a b) * n ^ (p a b) * (ε (b i * n) - ε n) := by
    calc (fun (n:ℕ) => q (b i * n) - q (r i n))
           ≤ᶠ[atTop] fun (n:ℕ) => ‖q (r i n) - q (b i * n)‖ := by
              filter_upwards with _; rw [norm_sub_rev]; exact le_norm_self _
         _ ≤ᶠ[atTop] fun (n:ℕ) => ‖(b i) ^ (p a b) * n ^ (p a b) * (ε (b i * n) - ε n)‖ :=
              h_main_norm
         _ =ᶠ[atTop] fun (n:ℕ) => (b i) ^ (p a b) * n ^ (p a b) * (ε (b i * n) - ε n) := by
              filter_upwards [eventually_gt_atTop ⌈(b i)⁻¹⌉₊, eventually_gt_atTop 1] with n hn hn'
              refine norm_of_nonneg ?_
              have h₁ := R.b_pos i
              have h₂ : 0 ≤ ε (b i * n) - ε n := by
                refine sub_nonneg_of_le <|
                  (strictAntiOn_smoothingFn.le_iff_le ?n_gt_one ?bn_gt_one).mpr ?le
                case n_gt_one =>
                  show 1 < (n:ℝ)
                  rw [Nat.one_lt_cast]
                  exact hn'
                case bn_gt_one =>
                  calc 1 = b i * (b i)⁻¹ := by rw [mul_inv_cancel (by positivity)]
                      _ ≤ b i * ⌈(b i)⁻¹⌉₊ := by gcongr; exact Nat.le_ceil _
                      _ < b i * n := by gcongr; rw [Nat.cast_lt]; exact hn
                case le => calc b i * n ≤ 1 * n := by have := R.b_lt_one i; gcongr
                                        _ = n := by rw [one_mul]
              positivity