Let $(X_n)_{n\geq 1}$ be a sequence of real-valued random variables.
I have to proof that if for every $\epsilon > 0: \sum_{n=1}^{\infty} \mathbb P(|X_n - X| > \epsilon) < \infty$ , then $X_n \to X$ almost sure.
Here's my attempt: To prove that $X_n$ converges a.s. I'd use Borel Cantelli lemma. Let $A_n = \{|X_n -X| > \epsilon\}$ and $\sum \epsilon^2 < \infty$ so by Borel Cantelli lemma, $\mathbb P(A_n$ occurs infinitely often) = 0, then for almost all $w$ for all $N$ greater than some index $N$ (depending on $w$), $|X_n(w) −X(w)| \leq \epsilon$, and thus $X_n \to X$ a.s.
Is this OK? Any hints / help / correction is appreciated.