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

Likewise, a sequence $v_0, v_1, v_2, \dots$ in a topological vector space $V$ is a Cauchy sequence if and only if for every neighborhood $U$ of $\mathbf{0}$ , 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.