Carefully state and prove the first and second Borel-Cantelli lemmas.

Now let $\left(A_{n}: n \in \mathbb{N}\right)$ be a sequence of events that are pairwise independent; that is, $\mathbb{P}\left(A_{n} \cap A_{m}\right)=\mathbb{P}\left(A_{n}\right) \mathbb{P}\left(A_{m}\right)$ whenever $m \neq n$. For $N \geqslant 1$, let $S_{N}=\sum_{n=1}^{N} 1_{A_{n}}$. Show that $\operatorname{Var}\left(S_{N}\right) \leqslant \mathbb{E}\left(S_{N}\right)$.

Using Chebyshev's inequality or otherwise, deduce that if $\sum_{n=1}^{\infty} \mathbb{P}\left(A_{n}\right)=\infty$, then $\lim _{N \rightarrow \infty} S_{N}=\infty$ almost surely. Conclude that $\mathbb{P}\left(A_{n}\right.$ infinitely often $)=1 .$