lipschitzOnWith_iff_dist_le_mul — Mathlib

English

The nth coordinate of Kuratowski embedding equals dist(a, x_n) − dist(x_0, x_n).

Русский

n-я координата встраивания Куротковского равна dist(a, x_n) − dist(x_0, x_n).

LaTeX

$$$(\mathrm{KuratowskiEmbedding}(x,a))_n = \operatorname{dist}(a, x_n) - \operatorname{dist}(x_0, x_n)$$$

Lean4

theorem lipschitzOnWith_iff_dist_le_mul [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0} {s : Set α} {f : α → β} :
    LipschitzOnWith K f s ↔ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ K * dist x y :=
  by
  simp only [LipschitzOnWith, edist_nndist, dist_nndist]
  norm_cast

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