We have real-valued random variables $\{X_n\}_{n=1}^\infty$, $\{Y_n\}_{n=1}^\infty$, $X$ and $Y$.
$X_n \rightarrow X$ in distribution and $Y_n \rightarrow Y$ in distribution, respectively. Also, $X$ and $Y$ are independent, as well as $X_n$ and $Y_n$ for any $n \ge1$.
Now, we would like to prove $X_n + Y_n \rightarrow X+Y$ in distribution. What I have in mind is to first prove $(X_n,Y_n) \rightarrow (X,Y)$ in distribution and then invoke the continuous mapping theorem.
To prove $(X_n,Y_n) \rightarrow (X,Y)$ in distribution, we only need to prove for every bounded Lipschitz function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$, we have $E[f(X_n,Y_n)] \rightarrow E[f(X,Y)]$ as $n \rightarrow \infty$, by the Portmanteau Theorem. Here's what I ran into problem: $$E[f(X_n,Y_n)] = \int_\mathbb{R} \left( \int_{\mathbb{R}} f(x,y) \mathcal{P}_{X_n}(dx) \right) \mathcal{P}_{Y_n}(dy),$$ where $\mathcal{P}_{X_n}, \mathcal{P}_{Y_n}$ are the distributions of $X_n$ and $Y_n$. How to prove $E[f(X_n,Y_n)] \rightarrow E[f(X,Y)]$ from this point?