TITLE: A sequence of random variables with bounded variance
QUESTION [5 upvotes]: If $\{X_n\}$ is a sequence of random variable with bounded variance:
  $$E|X_n^2|\le M<\infty,$$ and $X_n\to X$ in $L^1$, show that
  $$E|X^2|\le M.$$

I tried to use Fatou's lemma, Cauchy-Schwarz ... , but I failed to prove it. Can anyone give some hint?

REPLY [3 votes]: The following fact is well known (and strongly related to the Riesz-Fisher theorem):

If $X_n$ tends to $X$ in $L^1$, we can find a subsequence $X_{\phi(n)}$ such that $X_{\phi(n)}$ tends to $X$ both in $L^1$ and almost surely.

Applying Fatou's lemma to this subsequence yields
$$
E(|X|^2) = E(\liminf_{n\to\infty} |X_{\phi(n)}|^2) \leq \liminf_{n\to\infty} E(|X_{\phi(n)}|^2) \leq M.
$$
Therefore $E(|X|^2) \leq M$, and $E(|X|)^2 \leq M$ also holds because of Jensen's inequality (or Cauchy-Schwarz's if you prefer).