Let $E$ be a set, let $(Y,d_Y)$ be a [metric space](/page/Metric%20Space), let $(f_n)_{n=1}^{\infty}$ be a sequence of functions $f_n:E \to Y$, and let $f:E \to Y$ be a function. If $(f_n)_{n=1}^{\infty}$ converges uniformly to $f$ on $E$, then $(f_n)_{n=1}^{\infty}$ is uniformly Cauchy on $E$; that is, for every $\varepsilon>0$ there exists $N \in \mathbb{N}$ such that for all $m,n \ge N$ and all $x \in E$,

\begin{align*} d_Y(f_n(x),f_m(x))<\varepsilon. \end{align*}