Suppose $\displaystyle X_{n} \rightarrow X$ in probability and $\displaystyle X_{n}^{2} \le W, \forall n$ where $\displaystyle E(W)$ is finite. Show $\displaystyle X_{n} \rightarrow X$ in mean square.

So far I have:

$\displaystyle E((X_{n}-X)^2)=E(X_{n}^{2})-2E(X_{n}X)+E(X^{2})

\le E(W)-2E(X_{n}X)+E(X^{2})$

I'm not even sure if I'm on track or not. I can't think of how to use the convergence in probability to get anywhere.