frequently_zero_iff_eventually_zero

English

For an analytic f at z0, f is frequently zero in a punctured neighborhood iff f is eventually zero in a neighborhood of z0.

Русский

Для аналитической f в точке z0 частые нули в punctured окрестности эквивалентны существованию нулей в окрестности z0.

LaTeX

$$$(\exists^{\!}\; z \to z_0, f(z)=0) \iff (\forall^\! z \in (nhds z0), f(z)=0)$$$

Lean4

theorem frequently_zero_iff_eventually_zero {f : 𝕜 → E} {w : 𝕜} (hf : AnalyticAt 𝕜 f w) :
    (∃ᶠ z in 𝓝[≠] w, f z = 0) ↔ ∀ᶠ z in 𝓝 w, f z = 0 :=
  ⟨hf.eventually_eq_zero_or_eventually_ne_zero.resolve_right, fun h => (h.filter_mono nhdsWithin_le_nhds).frequently⟩

Похожие утверждения