Borel-Cantelli problem: Prove sum of probabilities is finite

Question

I am looking for good solutions to this problem. Could you please help me with this? Any solution would be appreciated.

Let $\mathit{X_n}$ be independent random variables. Show that if $\mathit{X_n}$ converges almost surely, then for any $\epsilon > 0$:

$$\sum_{n\ge1} \mathit{P}(|\mathit{X_n} - \mathit{X_{n+1}}| > \epsilon) < \infty$$