 Solution

The event that infinitely many $A_n$ occur is ${\mathrm{limsup}}_n{A}_n={\bigcap }_{m\ge 1}{\bigcup }_{n\ge m}{A}_n$. For every $m$, the union bound and monotonicity of a probability measure giveThe right-hand side tends to zero as the tail of a convergent series. Therefore the first Borel-Cantelli conclusion isThis proves the Borel-Cantelli first lemma without any independence assumption.