Let $X_n,Y_n,n\geq 1$ be sequences of real random variables such that $$ \int f(X_n)\,d\mathbb{P}-\int f(Y_n)\,d\mathbb{P}\to 0 $$ for all bounded and continuous functions $f$.
For which sets $A\subset\mathbb{R}$ does it hold that $$ \mathbb{P}(X_n\in A)-\mathbb{P}(Y_n\in A)\to 0 ? $$ I realize that I cannot use the Portmanteau theorem. Are there any additional assumptions that I would need for this to work?
What about the special case where $X_n=x_n+Z$ and $Y_n=y_n+Z$ with numerical sequences $x_n,y_n,n\geq 1$ with $x_n-y_n\to 0$?
Thank you