eventually_log_b_mul_pos — Mathlib

English

For each i, r_i(n) tends to a real value after casting, i.e., Tendsto (r_i(n)) as real toTop.

Русский

Для каждого i, r_i(n) сходится к вещественному значению после приведения к вещественному, Tendsto (r_i(n)) atTop atTop.

LaTeX

$$$\forall i,\; Tendsto (r_i(n))\; (atTop)\; (atTop)^{\mathbb{R}}$$$

Lean4

theorem eventually_log_b_mul_pos : ∀ᶠ (n : ℕ) in atTop, ∀ i, 0 < log (b i * n) :=
  by
  rw [Filter.eventually_all]
  intro i
  have h : Tendsto (fun (n : ℕ) => log (b i * n)) atTop atTop :=
    Tendsto.comp tendsto_log_atTop <| Tendsto.const_mul_atTop (b_pos R i) tendsto_natCast_atTop_atTop
  exact h.eventually_gt_atTop 0

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