Let $X_n, Y_n, X$ be real random variables such that $X_n \to X$ weakly and $\mathbb{P}_{Y_n} = N(0, 1/n)$ for all positive integers $n$.
I am trying to prove that $X_n + Y_n \to X$ weakly as well.
I have been trying to prove that $E[f(Y_n)] \to 0$ for all continuous bounded $f$, which would imply the statement. But i wasn't able to prove this, thought that this might be wrong approach.
I would be grateful for your hints or ideas. Thanks.