converges uniformly

Let $X$ be a set, $(Y,\rho)$ a metric space and $\{f_{n}\}$ a sequence of functions from $X$ to $Y$ , and $f\colon X\to Y$ another function.

If for every $\varepsilon>0$ there exists an integer $N$ such that

\rho(f_{n}(x),f(x))<\varepsilon

for all $x\in X$ and all $n>N$ , then we say that $f_{n}$ converges uniformly to $f$ .