English
In a pseudometric space, a sequence is Cauchy if and only if for every ε>0 there exists N such that for all m,n≥N we have dist(u_m,u_n)<ε.
Русский
В псевдометрическом пространстве последовательность Коши тогда и только тогда, когда для каждого ε>0 существует N such that для всех m,n≥N выполняется dist(u_m,u_n)<ε.
LaTeX
$$$\\displaystyle \\text{CauchySeq}(u) \\iff \\forall \\varepsilon>0, \\exists N, \\forall m\\ge N, \\forall n\\ge N,\\; \\operatorname{dist}(u_m,u_n) < \\varepsilon.$$$
Lean4
/-- In a pseudometric space, Cauchy sequences are characterized by the fact that, eventually,
the distance between its elements is arbitrarily small -/
theorem cauchySeq_iff {u : β → α} : CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε :=
uniformity_basis_dist.cauchySeq_iff
