 Borel-Cantelli first lemma

For arbitrary events $A_n$, if ${\sum }_n\mathbb{P}(A_n)<\infty$, then $\mathbb{P}({\mathrm{limsup}}_n{A}_n)=0$. The union bound gives $\mathbb{P}(\mathrm{limsup}{A}_n)\le {\sum }_{n\ge m}\mathbb{P}(A_n)\to 0$. No independence is required.