map_div_atTop_eq — Mathlib

English

For an integer exponent n with n > 0, the map x ↦ x^n tends to infinity along atTop.

Русский

При целочисленном показательe n > 0 отображение x^n растет до бесконечности при x → ∞.

LaTeX

$$$\forall n \in \mathbb{Z},\; n>0 \Rightarrow \operatorname{Tendsto}(x \mapsto x^{n})\ atTop\ atTop$$$

Lean4

theorem map_div_atTop_eq (k : α) (hk : 0 < k) : map (fun a => a / k) atTop = atTop :=
  map_atTop_eq_of_gc (fun b => k * b) 1 (fun _ _ h => div_le_div_of_nonneg_right h (le_of_lt hk))
    (fun a b _ => (by rw [div_le_iff₀' hk])) fun b _ => (by rw [mul_div_assoc, mul_div_cancel₀]; exact ne_of_gt hk)

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