Dominated stochastic processes with a common limit

Question

Let $A^n(t)$ and $B^n(t)$ be cadlag stochastic processes on some probability space satisfying $A^n(t)\leq B^n(t)$ for each $n$. Suppose that $A^n(t)$ and $B^n(t)$ each converge weakly (in the Skorokhod topology) to some common limit $L(t)$ as $n\rightarrow \infty$. Is it true that $B^n(t)-A^n(t)$ converges to zero? Any reference or suggestion would be much appreciated.