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

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

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