Let $S$ be a set, let $(Y,e)$ be a [metric space](/page/Metric%20Space), and let $B(S,Y)$ denote the set of bounded maps from $S$ to $Y$, equipped with the uniform metric $d_\infty$ defined by $d_\infty(g,h)=\sup_{s\in S} e(g(s),h(s))$ for $g,h\in B(S,Y)$. Let $f:S\to Y$ be an element of $B(S,Y)$, and let $(f_k)_{k\in\mathbb N}$ be a sequence in $B(S,Y)$. Then $f_k\to f$ in the metric space $(B(S,Y),d_\infty)$ if and only if $f_k$ converges uniformly to $f$ on $S$.