/-- If a monotone sequence `u` is such that `u n / n` tends to a limit `l` along subsequences with
exponential growth rate arbitrarily close to `1`, then `u n / n` tends to `l`. -/
theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ)
    (hmono : Monotone u)
    (hlim : ∀ a : ℝ, 1 < a → ∃ c : ℕ → ℕ, (∀ᶠ n in atTop, (c (n + 1) : ℝ) ≤ a * c n) ∧
      Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / c n) atTop (𝓝 l)) :
    Tendsto (fun n => u n / n) atTop (𝓝 l) := by
  /- To check the result up to some `ε > 0`, we use a sequence `c` for which the ratio
    `c (N+1) / c N` is bounded by `1 + ε`. Sandwiching a given `n` between two consecutive values of
    `c`, say `c N` and `c (N+1)`, one can then bound `u n / n` from above by `u (c N) / c (N - 1)`
    and from below by `u (c (N - 1)) / c N` (using that `u` is monotone), which are both comparable
    to the limit `l` up to `1 + ε`.
    We give a version of this proof by clearing out denominators first, to avoid discussing the sign
    of different quantities. -/
  have lnonneg : 0 ≤ l := by
    rcases hlim 2 one_lt_two with ⟨c, _, ctop, clim⟩
    have : Tendsto (fun n => u 0 / c n) atTop (𝓝 0) :=
      tendsto_const_nhds.div_atTop (tendsto_nat_cast_atTop_iff.2 ctop)
    apply le_of_tendsto_of_tendsto' this clim fun n => _
    simp_rw [div_eq_inv_mul]
    exact fun n => mul_le_mul_of_nonneg_left (hmono (zero_le _)) (inv_nonneg.2 (Nat.cast_nonneg _))
  have A : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, u n - n * l ≤ ε * (1 + ε + l) * n := by
    intro ε εpos
    rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
    have L : ∀ᶠ n in atTop, u (c n) - c n * l ≤ ε * c n := by
      rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
        Asymptotics.isLittleO_iff] at clim
      filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
      have cnpos : 0 < c n := cnpos'
      calc
        u (c n) - c n * l = (u (c n) / c n - l) * c n := by
          simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
        _ ≤ ε * c n := by
          refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
          exact le_trans (le_abs_self _) hn
    obtain ⟨a, ha⟩ :
      ∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
      eventually_atTop.1 (cgrowth.and L)
    let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
    filter_upwards [Ici_mem_atTop M] with n hn
    have exN : ∃ N, n < c N := by
      rcases (tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
      exact ⟨N, by linarith only [hN]⟩
    let N := Nat.find exN
    have ncN : n < c N := Nat.find_spec exN
    have aN : a + 1 ≤ N := by
      by_contra! h
      have cNM : c N ≤ M := by
        apply le_max'
        apply mem_image_of_mem
        exact mem_range.2 h
      exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
    have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN
    have cNn : c (N - 1) ≤ n := by
      have : N - 1 < N := Nat.pred_lt Npos.ne'
      simpa only [not_lt] using Nat.find_min exN this
    have IcN : (c N : ℝ) ≤ (1 + ε) * c (N - 1) := by
      have A : a ≤ N - 1 := by
        apply @Nat.le_of_add_le_add_right a 1 (N - 1)
        rw [Nat.sub_add_cancel Npos]
        exact aN
      have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
      have := (ha _ A).1
      rwa [B] at this
    calc
      u n - n * l ≤ u (c N) - c (N - 1) * l := by
        apply sub_le_sub (hmono ncN.le)
        apply mul_le_mul_of_nonneg_right (Nat.cast_le.2 cNn) lnonneg
      _ = u (c N) - c N * l + (c N - c (N - 1)) * l := by ring
      _ ≤ ε * c N + ε * c (N - 1) * l := by
        apply add_le_add
        · apply (ha _ _).2
          exact le_trans (by simp only [le_add_iff_nonneg_right, zero_le']) aN
        · apply mul_le_mul_of_nonneg_right _ lnonneg
          linarith only [IcN]
      _ ≤ ε * ((1 + ε) * c (N - 1)) + ε * c (N - 1) * l :=
        (add_le_add (mul_le_mul_of_nonneg_left IcN εpos.le) le_rfl)
      _ = ε * (1 + ε + l) * c (N - 1) := by ring
      _ ≤ ε * (1 + ε + l) * n := by
        refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
        apply mul_nonneg εpos.le
        linarith only [εpos, lnonneg]
  have B : ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (n : ℝ) * l - u n ≤ ε * (1 + l) * n := by
    intro ε εpos
    rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
    have L : ∀ᶠ n : ℕ in atTop, (c n : ℝ) * l - u (c n) ≤ ε * c n := by
      rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isLittleO_one_iff ℝ,
        Asymptotics.isLittleO_iff] at clim
      filter_upwards [clim εpos, ctop (Ioi_mem_atTop 0)] with n hn cnpos'
      have cnpos : 0 < c n := cnpos'
      calc
        (c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
          simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
        _ ≤ ε * c n := by
          refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
          simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
          exact le_trans (neg_le_abs_self _) hn
    obtain ⟨a, ha⟩ :
      ∃ a : ℕ,
        ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=
      eventually_atTop.1 (cgrowth.and L)
    let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
    filter_upwards [Ici_mem_atTop M] with n hn
    have exN : ∃ N, n < c N := by
      rcases (tendsto_atTop.1 ctop (n + 1)).exists with ⟨N, hN⟩
      exact ⟨N, by linarith only [hN]⟩
    let N := Nat.find exN
    have ncN : n < c N := Nat.find_spec exN
    have aN : a + 1 ≤ N := by
      by_contra! h
      have cNM : c N ≤ M := by
        apply le_max'
        

Proof state:
case H2
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
  ∀ (a : ℝ),
    1 < a →
      ∃ c,
        (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
          Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
ε : ℝ
εpos : 0 < ε
c : ℕ → ℕ
cgrowth : ∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ (1 + ε) * ↑(c n)
ctop : Tendsto c atTop atTop
clim : Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
L : ∀ᶠ (n : ℕ) in atTop, ↑(c n) * l - u (c n) ≤ ε * ↑(c n)
a : ℕ
ha : ∀ (b : ℕ), a ≤ b → ↑(c (b + 1)) ≤ (1 + ε) * ↑(c b) ∧ ↑(c b) * l - u (c b) ≤ ε * ↑(c b)
M : ℕ := max' (image (fun i => c i) (range (a + 1))) (_ : Finset.Nonempty (image (fun i => c i) (range (a + 1))))
n : ℕ
hn : n ∈ Set.Ici M
exN : ∃ N, n < c N
N : ℕ := Nat.find exN
ncN : n < c N
h : N < a + 1
⊢ c N ∈ image (fun i => c i) (range (a + 1))

Next tactic:
apply mem_image_of_mem