 Solution

The first Borel-Cantelli lemma says that ${\sum }_n\mathrm{Pr}(A_n)<\infty$ implies $\mathrm{Pr}(A_n\text{ infinitely often})=0$, without an independence assumption. Indeed, for every $N$,The decreasing intersection of these tail unions is the infinitely-often event; continuity of probability from above proves the assertion. The second lemma says that for independent $A_n$, divergence of the probability sum implies $\mathrm{Pr}(A_n\text{ infinitely often})=1$.