Let $X_n,Y_n$ be sequences of random variables defined on some common probability space. Suppose $X_n\leq Y_n$ (a.s.) and $X_n\Rightarrow L$, $Y_n\Rightarrow L$ as $n\rightarrow\infty$, for some distribution $L$. Is it true that $Y_n-X_n\Rightarrow 0$?
If we have joint convergence $(X_n,Y_n)\Rightarrow (L_1,L_2)$ with each $L_i$ having marginal distribution $L$, the claim follows from an application of the Skorohod representation theorem. I'm not sure about the claim without this condition.