$(X_n ; n>0)$ gaussian random variable sequence, converges uniformly, implies that $X_n$ converges in $L^2$

Question

Let $(X_n ; n>0)$ a gaussian random variables sequence. $X_n$ converges to $X$ Uniformaly ie $\sup\limits_{\omega\in\Omega}|X_n-X|$ converges to $0$. We Must prove that : $$E((X_n-X)^2)$$ converges to $0$.

Answer 1

Using Bounded Convergence Theorem,

$$E[|X_n-X|^2]\leq \Big(\sup_{\Omega }|X_n-X|\Big)^2.$$