Let $X_1,X_2,...$ and $Y_1,Y_2,...$ be two sequences of random variables such that $\mathcal{L}(X_n)\xrightarrow{w}\mathcal{L}(X)$ and $|Y_n-X_n|\xrightarrow{w}0$ as $n\rightarrow\infty$ Show that $$\mathcal{L}(Y_n)\xrightarrow{w}\mathcal{L}(X)$$ as $n\rightarrow\infty$.
I have the next attempt:
I know that convergence in probability implies convergence weakly or in distribution, then given $\epsilon>0$ I can say that:
$$|Y_n-X_n|<\epsilon$$
Also since $\mathcal{L}(X_n)\xrightarrow{w}\mathcal{L}(X)$, I can say that $Ef(X_n)\rightarrow Ef(X)$ for a bounded Lipschitz function using the Portamnteau theorem.
I don't know how to use these facts to conclude.