approx_of_mem_C — Mathlib

English

The mapping from unit interval to NNReal is continuous when viewed with their natural topologies.

Русский

Отображение from единичного интервала в NNReal непрерывно в естественных топологиях.

LaTeX

$$$Continuous(toNNReal)$$$

Lean4

theorem approx_of_mem_C (c : CU P) (n : ℕ) {x : X} (hx : x ∈ c.C) : c.approx n x = 0 := by
  induction n generalizing c with
  | zero => exact indicator_of_notMem (fun (hU : x ∈ c.Uᶜ) => hU <| c.subset hx) _
  | succ n ihn =>
    simp only [approx]
    rw [ihn, ihn, midpoint_self]
    exacts [c.subset_right_C hx, hx]

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