Let $X$ be any set, and let $(Y,d)$ be a metric space. A sequence $f_{1},f_{2},\dots$ of functions mapping $X$ to $Y$ is said to be uniformly convergent to another function $f$ if, for each $\varepsilon>0$, there exists $N$ such that, for all $x$ and all $n>N$, we have $d(f_{n}(x),f(x))<\varepsilon$. This is denoted by $f_{n}\xrightarrow[]{u}f$, or “$f_{n}\rightarrow f$ uniformly” or, less frequently, by $f_{n}\rightrightarrows f$.