Let $E$ be a finite set, let $(Y,d_Y)$ be a [metric space](/page/Metric%20Space), and let $(f_n)_{n=1}^{\infty}$ be a sequence of functions $f_n:E \to Y$. Suppose that for every $x \in E$, the sequence $(f_n(x))_{n=1}^{\infty}$ is Cauchy in $(Y,d_Y)$. Then $(f_n)_{n=1}^{\infty}$ is uniformly Cauchy on $E$, meaning that 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_m(x),f_n(x))<\varepsilon. \end{align*}