weak convergence and composition

Question

Assume $X_n$ is a sequence of random variables defined on a common probability space and $X_n$ converges weakly (in distribution) to $X$ as $n \to \infty$. Assume $u_n$ is a sequence of integer valued random variables defined on the same probability space as the $X_n$, and satisfy $\lim_{n \to \infty} u_n =\infty$ almost everywhere. Is it true that $X_{u_n}$ converges weakly to $X$ as $n \to \infty$?