Let $\mathscr{F}$ be a Borel field and $\{E_n\}_{n\geq1}\in\mathscr{F}$ are events. We have the first Borel-Cantelli Lemma $$\sum_{n=1}^\infty \mathscr{P}\{E_n\} < \infty \implies \mathscr{P}\{E_n\mbox{ i.o.}\}=0,$$but, the converse is NOT true.
$\bullet$ Counterexample.
We need to find events $\{E_n\}$ satisfy $\mathscr{P}\{E_n\mbox{ i.o.}\}=0$ but $\sum_n \mathscr{P}\{E_n\} = \infty.$
Let $\Omega=[0,1]$ and $\mathscr{F}=\mathscr{B}[0,1]$. Define the events and the corresponding probabilities $$A_n=\left[0,\frac{1}{n}\right]\mbox{ and }\mathscr{P}\{A_n\}=\frac{1}{n},\;n=1,2,\ldots$$Since $A_1\supset A_2\supset\cdots, $ $$\{A_n\mbox{ i.o.}\}=\bigcap_{n=1}^\infty\bigcup_{m=n}^\infty A_m=\bigcap_{n=1}^\infty A_n=\emptyset.$$Thus, $\mathscr{P}\{A_n\mbox{ i.o.}\}=0$.
But, $$\sum_{n=1}^\infty \mathscr{P}\{A_n\}=\sum_{n=1}^\infty\frac{1}{n}=\infty.$$
$\Box$
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