Does $L^2$ convergence implies convergence of the second moments?

Question

This is a simple question related to $L^p$ convergence.

Let us assume that $X_n$ converge to $X$ in $L^2$, could we get that $E(X_n^2)$ converge to $E(X^2)$?

I kind of think it is not true, but cannot find an counter example.

Answer 1

Yes, this is true.

$$ \begin{align} |\mathbb{E}[X_n^2] - \mathbb{E}[X^2]| &= |\mathbb{E}[(X_n-X)^2] - 2\mathbb{E}[X(X-X_n)]| \\ &\leq \mathbb{E}[(X_n-X)^2] + 2\sqrt{\mathbb{E}[X^2]\mathbb{E}[(X-X_n)^2]} \end{align}$$