le_iInf — Mathlib

English

If all i satisfy f(i) ≤ a for some a ≥ 0, then ⨆ i, f(i) ≤ a.

Русский

Если для всех i выполнено f(i) ≤ a и a ≥ 0, то ⨆ i, f(i) ≤ a.

LaTeX

$$$\\forall i, f(i) \\le a \\quad\\land\\; 0 \\le a \\Rightarrow \\sup_{i} f(i) \\le a$$$

Lean4

/-- As `⨅ i, f i = 0` when the domain of the real-valued function `f` is empty, it suffices to show
that all values of `f` are at least some nonpositive number `a` to show that `a ≤ ⨅ i, f i`.

See also `le_ciInf`. -/
protected theorem le_iInf (hf : ∀ i, a ≤ f i) (ha : a ≤ 0) : a ≤ ⨅ i, f i :=
  Real.le_sInf (Set.forall_mem_range.2 hf) ha

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