Cauchy sequence

A sequence $x_0,x_1,x_2,\mathrm{\dots }$ in a metric space $(X,d)$ is a Cauchy sequence if, for every real number $ϵ>0$, there exists a natural number $N$ such that whenever $n,m>N$.

Likewise, a sequence $v_0,v_1,v_2,\mathrm{\dots }$ in a topological vector space $V$ is a Cauchy sequence if and only if for every neighborhood $U$ of $\mathrm{𝟎}$, there exists a natural number $N$ such that $v_n-v_m\in U$ for all $n,m>N$. These two definitions are equivalent when the topology of $V$ is induced by a metric.