I have a problem with the following exercise.
Let $(X_n)$ be a limited martingale (there exists $M>0$ such that for each $n$ is valid $\mathbb{P}(|X_n|<M)=1$). I have to prove that $X_n$ converges in $L^2$.
I know that for each $m>n$ is valid $\mathbb{E}\left[\left(X_m-X_n\right)^2\right] = \operatorname{Var}(X_m)-\operatorname{Var}(X_n)$ but I can not go on.
Thanks in advance for your help.