isEquivalent_nat_floor — Mathlib

English

The floor map is asymptotically equivalent to the identity on large arguments in atTop; floor grows like x.

Русский

Округление вниз по перспективе эквивалентности к идентичности на больших аргументах; floor x растёт как x.

LaTeX

$$$\mathrm{isEquivalent}(\, x \mapsto \lfloor x \rfloor) \sim (\, x \mapsto x\,)$$$

Lean4

theorem isEquivalent_nat_floor : (fun (x : R) ↦ ↑⌊x⌋₊) ~[atTop] (fun x ↦ x) :=
  by
  refine isEquivalent_of_tendsto_one ?_ tendsto_nat_floor_div_atTop
  filter_upwards with x hx using by rw [hx, Nat.floor_zero, Nat.cast_eq_zero]

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