Levy's 0-1 law and Martinagle
Solution 1:
$X_n$ is a martingale, and so converges (boundedly) to a random variable $X_\infty$. You'll be able to write a recursion for the second moments $\Bbb E[X_n^2]$, leading, after a passage to the limit as $n\to\infty$, to the conclusion that $\Bbb E[X_\infty^2] = \Bbb E[X_\infty]$.