English
In a real or metric setting, similarity between two families is controlled by pairwise distances: if vectors are similar, then all pairwise distances are scaled equivalently.
Русский
Если две системы вменяемы как подобные, то попарные расстояния между их элементами пропорциональны друг другу.
LaTeX
$$$\\text{Similar } v_1 v_2 \\iff \\exists r \\ge 0, r \\neq 0 \\land \\text{Pairwise } (i_1,i_2) \\mapsto \\operatorname{edist}(v_1(i_1),v_1(i_2)) = r \\cdot \\operatorname{edist}(v_2(i_1),v_2(i_2)).$$$
Lean4
/-- A version of **Bolzano-Weierstrass**: in a proper metric space (e.g. $ℝ^n$),
every bounded sequence has a converging subsequence. This version assumes only
that the sequence is frequently in some bounded set. -/
theorem tendsto_subseq_of_frequently_bounded (hs : IsBounded s) {x : ℕ → X} (hx : ∃ᶠ n in atTop, x n ∈ s) :
∃ a ∈ closure s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (x ∘ φ) atTop (𝓝 a) :=
have hcs : IsSeqCompact (closure s) := hs.isCompact_closure.isSeqCompact
have hu' : ∃ᶠ n in atTop, x n ∈ closure s := hx.mono fun _n hn => subset_closure hn
hcs.subseq_of_frequently_in hu'
