le_ciSup — Mathlib

English

Let ι be finite and f: ι → α. Then for every i ∈ ι, we have f(i) ≤ ⨆ j, f(j).

Русский

Пусть ι конечен и есть функция f: ι → α. Тогда для каждого i ∈ ι выполняется f(i) ≤ ⨆_j f(j).

LaTeX

$$$\forall i,\ f(i) \le \big\langle\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n" + "?"$$

Lean4

theorem le_ciSup (i : ι) : f i ≤ ⨆ j, f j :=
  by
  suffices BddAbove (range f) from _root_.le_ciSup this i
  let : Fintype ι := Fintype.ofFinite ι
  use Finset.sup' Finset.univ ⟨i, Finset.mem_univ i⟩ f
  simp only [mem_upperBounds, mem_range, forall_exists_index, forall_apply_eq_imp_iff]
  exact fun j ↦ Finset.le_sup' f <| Finset.mem_univ j

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