lemma eventually_atTop_sumTransform_le :
    ∃ c > 0, ∀ᶠ (n:ℕ) in atTop, ∀ i, sumTransform (p a b) g (r i n) n ≤ c * g n := by
  obtain ⟨c₁, hc₁_mem, hc₁⟩ := R.exists_eventually_const_mul_le_r
  obtain ⟨c₂, hc₂_mem, hc₂⟩ := R.g_grows_poly.eventually_atTop_le_nat hc₁_mem
  have hc₁_pos : 0 < c₁ := hc₁_mem.1
  refine ⟨max c₂ (c₂ / c₁ ^ (p a b + 1)), by positivity, ?_⟩
  filter_upwards [hc₁, hc₂, R.eventually_r_pos, R.eventually_r_lt_n, eventually_gt_atTop 0]
    with n hn₁ hn₂ hrpos hr_lt_n hn_pos
  intro i
  have hrpos_i := hrpos i
  have g_nonneg : 0 ≤ g n := R.g_nonneg n (by positivity)
  rcases le_or_lt 0 (p a b + 1) with hp|hp
  case h.inl =>   -- 0 ≤ p a b + 1
    calc sumTransform (p a b) g (r i n) n
           = n ^ (p a b) * (∑ u in Finset.Ico (r i n) n, g u / u ^ ((p a b) + 1)) := by rfl
         _ ≤ n ^ (p a b) * (∑ u in Finset.Ico (r i n) n, c₂ * g n / u ^ ((p a b) + 1)) := by
                gcongr with u hu
                rw [Finset.mem_Ico] at hu
                have hu' : u ∈ Set.Icc (r i n) n := ⟨hu.1, by linarith⟩
                refine hn₂ u ?_
                rw [Set.mem_Icc]
                refine ⟨?_, by norm_cast; linarith⟩
                calc c₁ * n ≤ r i n := by exact hn₁ i
                          _ ≤ u := by exact_mod_cast hu'.1
         _ ≤ n ^ (p a b) * (∑ _u in Finset.Ico (r i n) n, c₂ * g n / (r i n) ^ ((p a b) + 1)) := by
                  gcongr with u hu; rw [Finset.mem_Ico] at hu; exact hu.1
         _ ≤ n ^ (p a b) * (Finset.Ico (r i n) n).card • (c₂ * g n / (r i n) ^ ((p a b) + 1)) := by
                  gcongr; exact Finset.sum_le_card_nsmul _ _ _ (fun x _ => by rfl)
         _ = n ^ (p a b) * (Finset.Ico (r i n) n).card * (c₂ * g n / (r i n) ^ ((p a b) + 1)) := by
                  rw [nsmul_eq_mul, mul_assoc]
         _ = n ^ (p a b) * (n - r i n) * (c₂ * g n / (r i n) ^ ((p a b) + 1)) := by
                  congr; rw [Nat.card_Ico, Nat.cast_sub (le_of_lt <| hr_lt_n i)]
         _ ≤ n ^ (p a b) * n * (c₂ * g n / (r i n) ^ ((p a b) + 1)) := by
                  gcongr; simp only [tsub_le_iff_right, le_add_iff_nonneg_right, Nat.cast_nonneg]
         _ ≤ n ^ (p a b) * n * (c₂ * g n / (c₁ * n) ^ ((p a b) + 1)) := by
                gcongr; exact hn₁ i
         _ = c₂ * g n * n ^ ((p a b) + 1) / (c₁ * n) ^ ((p a b) + 1) := by
                rw [← Real.rpow_add_one (by positivity) (p a b)]; ring
         _ = c₂ * g n * n ^ ((p a b) + 1) / (n ^ ((p a b) + 1) * c₁ ^ ((p a b) + 1)) := by
                rw [mul_comm c₁, Real.mul_rpow (by positivity) (by positivity)]
         _ = c₂ * g n * (n ^ ((p a b) + 1) / (n ^ ((p a b) + 1))) / c₁ ^ ((p a b) + 1) := by ring
         _ = c₂ * g n / c₁ ^ ((p a b) + 1) := by rw [div_self (by positivity), mul_one]
         _ = (c₂ / c₁ ^ ((p a b) + 1)) * g n := by ring
         _ ≤ max c₂ (c₂ / c₁ ^ ((p a b) + 1)) * g n := by gcongr; exact le_max_right _ _
  case h.inr =>   -- p a b + 1 < 0
    calc sumTransform (p a b) g (r i n) n
           = n ^ (p a b) * (∑ u in Finset.Ico (r i n) n, g u / u ^ ((p a b) + 1)) := by rfl
         _ ≤ n ^ (p a b) * (∑ u in Finset.Ico (r i n) n, c₂ * g n / u ^ ((p a b) + 1)) := by
                gcongr with u hu
                rw [Finset.mem_Ico] at hu
                have hu' : u ∈ Set.Icc (r i n) n := ⟨hu.1, by linarith⟩
                refine hn₂ u ?_
                rw [Set.mem_Icc]
                refine ⟨?_, by norm_cast; linarith⟩
                calc c₁ * n ≤ r i n      := by exact hn₁ i
                          _ ≤ u          := by exact_mod_cast hu'.1
         _ ≤ n ^ (p a b) * (∑ _u in Finset.Ico (r i n) n, c₂ * g n / n ^ ((p a b) + 1)) := by
                gcongr n ^ (p a b) * (Finset.Ico (r i n) n).sum (fun _ => c₂ * g n / ?_) with u hu
                rw [Finset.mem_Ico] at hu
                have : 0 < u := calc
                  0 < r i n := by exact hrpos_i
                  _ ≤ u := by exact hu.1
                exact rpow_le_rpow_of_exponent_nonpos (by positivity)
                  (by exact_mod_cast (le_of_lt hu.2)) (le_of_lt hp)
         _ ≤ n ^ (p a b) * (Finset.Ico (r i n) n).card • (c₂ * g n / n ^ ((p a b) + 1)) := by
                  gcongr; 

Proof state:
case h
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
c₁ : ℝ
hc₁_mem : c₁ ∈ Set.Ioo 0 1
hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
c₂ : ℝ
hc₂_mem : c₂ > 0
hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, g u ≤ c₂ * g ↑n
hc₁_pos : 0 < c₁
n : ℕ
hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)
hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, g u ≤ c₂ * g ↑n
hrpos : ∀ (i : α), 0 < r i n
hr_lt_n : ∀ (i : α), r i n < n
hn_pos : 0 < n
i : α
hrpos_i : 0 < r i n
g_nonneg : 0 ≤ g ↑n
hp : p a b + 1 < 0
⊢ ∑ _u in Ico (r i n) n, c₂ * g ↑n / ↑n ^ (p a b + 1) ≤ card (Ico (r i n) n) • (c₂ * g ↑n / ↑n ^ (p a b + 1))

Next tactic:
exact Finset.sum_le_card_nsmul _ _ _ (fun x _ => by rfl)