| import Mathlib.Analysis.SpecificLimits.Normed | |
| namespace OAI | |
| open Filter Topology | |
| namespace PiExponent | |
| theorem tendsto_nat_add_one_div_pow {r : ℝ} (hr : 1 < r) : | |
| Tendsto (fun n : ℕ => ((n : ℝ) + 1) / r ^ n) atTop (𝓝 0) := by | |
| have h₁ := tendsto_pow_const_div_const_pow_of_one_lt 1 hr | |
| have h₀ := tendsto_pow_const_div_const_pow_of_one_lt 0 hr | |
| simpa only [pow_one, pow_zero, ← add_div, zero_add] using h₁.add h₀ | |
| theorem tendsto_pow_div_nat_add_one {r : ℝ} (hr : 1 < r) : | |
| Tendsto (fun n : ℕ => r ^ n / ((n : ℝ) + 1)) atTop atTop := by | |
| have hpos : ∀ᶠ n : ℕ in atTop, 0 < ((n : ℝ) + 1) / r ^ n := | |
| Filter.Eventually.of_forall fun n => | |
| div_pos (by positivity) (pow_pos (zero_lt_one.trans hr) n) | |
| have hwithin : Tendsto (fun n : ℕ => ((n : ℝ) + 1) / r ^ n) | |
| atTop (𝓝[>] (0 : ℝ)) := | |
| tendsto_nhdsWithin_iff.mpr ⟨tendsto_nat_add_one_div_pow hr, hpos⟩ | |
| have hinv := hwithin.inv_tendsto_nhdsGT_zero | |
| change Tendsto (fun n : ℕ => (((n : ℝ) + 1) / r ^ n)⁻¹) atTop atTop at hinv | |
| simpa only [inv_div] using hinv | |
| end PiExponent | |
| end OAI | |