$\sum_{n=1}^{\infty} E|X_n - X| < \infty$ imples $X_n$ converges to $X$ almost surely
I'm sure this is an obvious one line proof and I'm being stupid here, but I cannot see how to show the above. I can see that $\sum_{n=1}^{\infty} P(|X_n - X|>\epsilon) < \infty$ would give the result via the Borel Cantelli lemma, am not sure how I can use that fact. Could I maybe approximate $|X_n - X|$ by simple functions? Then turn the expected value operator into a probability.