## Convegence in distribution

Let $X_n$ and $X$ random variables taing values in the metric space $(S,d)$.

The sequence $(X_n)_n$ is convergent to $X$ in distribution if
$E[f(X_n)] \to E[f(X)]$ for all $f:S\to R$ continuous and bounded.

I read somewhere that it's equivalent to consider only uniformly continuous and bounded $f$.
Could you give me a proof of this?