pos_mul_atTop — Mathlib

English

As above, a subgroup is dense in G if and only if it is not equal to zpowers(a) for any a ∈ G.

Русский

Как выше, подгруппа плотна в G тогда, когда она не равна zpowers(a) для любого a ∈ G.

LaTeX

$$$\overline{S} = G \iff \forall a\in G,\ S \neq \mathrm{zpowers}(a).$$$

Lean4

/-- In a linearly ordered semifield with the order topology, if `f` tends to a positive constant `C`
and `g` tends to `Filter.atTop` then `f * g` tends to `Filter.atTop`. -/
theorem pos_mul_atTop {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l (𝓝 C)) (hg : Tendsto g l atTop) :
    Tendsto (fun x => f x * g x) l atTop := by simpa only [mul_comm] using hg.atTop_mul_pos hC hf

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