English
In a nontrivial densely ordered archimedean topological multiplicative group G, a subgroup is either dense in G or is a cyclic subgroup, but not both; more precisely, for s ≤ G, either s is dense in G or s equals zpowers generated by some a, but not both.
Русский
В ненулевой densely упорядоченной архимедовой топологической мультипликативной группе G подгруппа либо плотна, либо циклична, но не одновременно.
LaTeX
$$$(\text{Dense}(s)) \oplus (\exists a, s = \mathrm{zpowers}(a)).$$$
Lean4
/-- In a linearly ordered semifield with the order topology, if `f` tends to `Filter.atTop` and `g`
tends to a positive constant `C` then `f * g` tends to `Filter.atTop`. -/
theorem atTop_mul_pos {C : 𝕜} (hC : 0 < C) (hf : Tendsto f l atTop) (hg : Tendsto g l (𝓝 C)) :
Tendsto (fun x => f x * g x) l atTop :=
by
refine tendsto_atTop_mono' _ ?_ (hf.atTop_mul_const (half_pos hC))
filter_upwards [hg.eventually (lt_mem_nhds (half_lt_self hC)), hf.eventually_ge_atTop 0] with x hg hf using
mul_le_mul_of_nonneg_left hg.le hf
