I tried proving the following lemma: If $(A_n)_n$ is a sequence of events in a probability space s.t $\sum_{n=1}^{\infty}\mathbb{P}[A_n]<\infty$, then $\mathbb{P}[\text{limsup}A_n]=0$.
My attempt: Since $\sum_{n=1}^{\infty}\mathbb{P}[A_n]<\infty$, we know that $\mathbb{P}[A_n]$ converges to $0$ as a sequence of numbers. So $\text{limsup}\mathbb{P}[A_n]=0$, so Fatou's lemma finishes the proof. Is my proof correct? I think the real question is, does $\sum_{n=1}^{\infty}\mathbb{P}[A_n]<\infty$ imply convergence of the series?