State and prove the first Borel-Cantelli Lemma.

Suppose that $\left(F_{n}\right)$ is a sequence of events in a common probability space such that $\mathbb{P}\left(F_{i} \cap F_{j}\right) \leqslant \mathbb{P}\left(F_{i}\right) \cdot \mathbb{P}\left(F_{j}\right)$ whenever $i \neq j$ and that $\sum_{n} \mathbb{P}\left(F_{n}\right)=\infty$.

Let $1_{F_{n}}$ be the indicator function of $F_{n}$ and let

$$S_{n}=\sum_{k \leqslant n} 1_{F_{k}} ; \mu_{n}=\mathbb{E}\left(S_{n}\right)$$

Use Chebyshev's inequality to show that

$$\mathbb{P}\left(S_{n}<\frac{1}{2} \mu_{n}\right) \leqslant \mathbb{P}\left(\left|S_{n}-\mu_{n}\right|>\frac{1}{2} \mu_{n}\right) \leqslant \frac{4}{\mu_{n}}$$

Deduce, using the first Borel-Cantelli Lemma, that $\mathbb{P}\left(F_{n}\right.$ infinitely often $)=1$.