Suppose $X_{n}$ is series of random variables such that it is $\sum_{n=1}^{\infty} E(X_{n}^{4}) < \infty$. Prove that $X_{n} \rightarrow 0$ a.s.

Question

Suppose $X_{n}$ is series of random variables such that it is $\sum_{n=1}^{\infty} E(X_{n}^{4}) < \infty$. According to Markov's inequality we know that $$P(\lvert X_{n} \rvert > \epsilon) \leq \frac{1}{\epsilon^{4}} \cdot E(\lvert X_{n} \rvert ^{4}).$$ Using the assumption we have: $$\sum_{n=1}^{\infty} P(\lvert X_{n} \rvert > \epsilon)\ \leq \frac{1}{\epsilon^{4}} \cdot \sum_{n=1}^{\infty} E(\lvert X_{n} \rvert ^{4}).$$ Therefore, $\sum_{n=1}^{\infty} P(\lvert X_{n} \rvert > \epsilon)\ < \infty$ which implies $X_{n} \rightarrow 0$ a.s. I hope that proof is correct :)