When we have two independent sequences or random variables $\{X_{n}\}$ and $\{Y_{n}\}$ for which $X_{n}$ converges weakly to $X$ ( $X_n \overset{w}{\rightarrow}X$) and $Y_n \overset{w}{\rightarrow}Y$ I want to show that holds $(X_n,Y_n) \overset{w}{\rightarrow}(X,Y)$
I know that holds for $f\in C_{b}$: $\mathbb{E}_{X}f(X_n)\rightarrow \mathbb{E}f(X)$ and the same for $Y$. I want to show that $\mathbb{E}_{X\times Y}f(X_n,Y_n)\rightarrow \mathbb{E}_{X\times Y}f(X,Y)$
As $X_n$ and $Y_n$ are independent I can write $\mathbb{E}_{X\times Y} = \mathbb{E}_X \times \mathbb{E}_Y$, the product measure.
So $\mathbb{E}_{X\times Y}f(X_n,Y_n)=\mathbb{E}_X (X_n,Y_n)\times \mathbb{E}_Y (X_n,Y_n)=\mathbb{E}_{X}(X,Y_{n})\times \mathbb{E}_Y (X_n,Y)$
I now need some help making the last step to get $\mathbb{E}_{X\times Y}(X,Y)$. Could anyone help me with this? I'd really prefer to do it without using characteristic functions if possible.