Let $X^n\sim Y^n$ for all $n$ and $X^n\to X$ and $Y^n \to Y$ both in probability. Is $X\sim Y$?
If all variables take values in some measurable space $(S,\mathbb S)$: I'm thinking $P(X\in A)=\lim \, P(X^n\in A) = \lim \, P(Y^n\in A) = P(Y \in A) $ for $A\in \mathbb S$. But suddenly i had doubts that convergence in probability implies that $\lim \, P(X^n\in A)=P(X\in A)$. Is the statement true? Is my argument?
Edit: Sorry. I was a bit fast there. Let's assume they take value in some metric space.