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a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | 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
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