I'm trying to prove the following statement:
Let $ X_n, X_0 $ be such R.V.'s that $ X_n $ converge to $ X_0 $ in distribution (weakly). Prove that there exist $ Y_n, Y_0 $ on the probabilistic space of an interval $(0,1)$ with $ \sigma$ - field of Borel sets and probabilistic measure given by Lebesgue measure such that for all $ i = 0,1\dots$ $ X_i, Y_i $ have the same distribution and $ Y_n(\omega) \rightarrow Y(\omega) $ for all $ \omega \in (0,1) $
I don't know where to begin. I know for each bounded, continuous function I have $ \int_\mathbb{R} f(x) \mu_{X_n}(dx) \rightarrow \int_\mathbb{R} f(x) \mu_{X_0}(dx) $, but where does that lead?