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

The sequence $\displaystyle (X_n)_n$ isconvergentto $\displaystyle X$in distributionif

$\displaystyle E[f(X_n)] \to E[f(X)]$ for all $\displaystyle f:S\to R$ continuous and bounded.

I read somewhere that it's equivalent to consider onlyuniformly continuousand bounded $\displaystyle f$.

Could you give me a proof of this?