Question:
My approach:
Fix any $x$ and $y$ in the codomains of $X$ and $Y$ respectively.
$\mathbb{P}(X_n \leq x, Y_n \leq y) = \mathbb{P}(X_n \leq x)\mathbb{P}(Y_n \leq y) \rightarrow \mathbb{P}(X \leq x)\mathbb{P}(Y \leq y)$, by the fact that $\mathbb{P}(X_n \leq x) \rightarrow \mathbb{P}(X \leq x)$ and $\mathbb{P}(Y_n \leq y) \rightarrow \mathbb{P}(Y \leq y)$ and by the product rule of convergence of sequence of real numbers. Using the three necessary and sufficient conditions of a function being a CDF, $\mathbb{P}(X \leq x)\mathbb{P}(Y \leq y)$ is a CDF, which is same as that of the random vector $(X,Y)$ constructed taking $X$ and $Y$ as independent.
Is there anything I'm missing? The official solutions given are all much longer and involve Portmanteau Lemma.