I know a theorem called Polya's theorem:
$X_n \rightarrow X$ in distribution as $n\rightarrow \infty$ is equivalent to $\sup_n | F_n(x) -F(x)| \rightarrow 0$ as $n \rightarrow \infty$, where $F_n, F$ are distribution functions of $X_n$ and $X$, respectively.
Do you know where I can find out the proof for this theorem? or do you have hints to prove it?
The supremum should be over all $x$, not over all $n$, it's pointless.
This statement fails in general without assumption that the limiting function is continuous. Counterexamples are quite obvious. Say, a sequence $X_n$ with CDF $F_n(x)=x^n1_{(0,1)}$ converges to $X=1$ in distribution, but $\sup_x | F_n(x) -F(x)|=1$.
You can see a proof of correct statement here: theorem 9 on page 5.