Let $(X\_n)$ be a sequence of random variables. If $\sum\_{n \geqslant 1} \mathbb{P}\left(\left|X\_{n}\right|>\varepsilon\right)<\infty$ for each $\varepsilon>0$, prove $\mathbb{P}\left(\lim \_{n \rightarrow \infty} X\_{n}=0\right)=1$.

I have already tried this several times. Here's my last and most improved attempt.

\begin{align\*}
 \sum\_{n \geqslant 1} \mathbb{P}\left(\left|X\_{n}\right|>\varepsilon\right)<\infty,\text{ }\forall\varepsilon>0\implies&\mathbb{P}(\left[|X\_{n}|>\varepsilon\right]\text{ i.o.})=0\\
 \implies&X\_{n}\overset{\underset{a.s.}{}}{\rightarrow}0\\
 \iff&\mathbb{P}\left(\lim \_{n \rightarrow \infty} X\_{n}=0\right)=1.
 \end{align\*}