tendsto_atTop_of_linearGrowthInf_natCast_pos

English

For a<b in Real with hb≥0, there are eventually natural numbers between a x and b x.

Русский

Для a<b в Real с hb≥0, неотражаемо существуют натуральные числа между a x и b x.

LaTeX

$$$\forall a

Lean4

theorem tendsto_atTop_of_linearGrowthInf_natCast_pos (h : (linearGrowthInf fun n ↦ v n : EReal) ≠ 0) :
    Tendsto v atTop atTop := by
  refine tendsto_atTop.2 fun M ↦ ?_
  have := tendsto_atTop_of_linearGrowthInf_pos (h.lt_of_le' (linearGrowthInf_natCast_nonneg v))
  refine (tendsto_nhds_top_iff_real.1 this M).mono fun n ↦ ?_
  rw [coe_coe_eq_natCast, Nat.cast_lt]
  exact le_of_lt

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