(a) Prove Lemma 3, the inclusion-exclusion principle.
(b) Prove that, when $\ell$ is odd,
$$
\begin{aligned}
\operatorname{Pr}\left(\bigcup_{i=1}^{n} E_{i}\right) \leq & \sum_{i=1}^{n} \operatorname{Pr}\left(E_{i}\right)-\sum_{i<j} \operatorname{Pr}\left(E_{i} \cap E_{j}\right) \\
&+\sum_{i<j<k} \operatorname{Pr}\left(E_{i} \cap E_{j} \cap E_{k}\right) \\
&-\cdots+(-1)^{\ell+1} \sum_{i_{1}<_{i}<\cdots<i \ell} \operatorname{Pr}\left(E_{i_{1}} \cap \cdots \cap E_{i_{\ell}}\right)
\end{aligned}
$$
(c) Prove that, when $\ell$ is even,
$$
\begin{aligned}
\operatorname{Pr}\left(\bigcup_{i=1}^{n} E_{i}\right) \geq \sum_{i=1}^{n} \operatorname{Pr}\left(E_{i}\right)-\sum_{i<j} \operatorname{Pr}\left(E_{i} \cap E_{j}\right) & \\
&+\sum_{i<j<k} \operatorname{Pr}\left(E_{i} \cap E_{j} \cap E_{k}\right) \\
&-\cdots+(-1)^{\ell+1} \sum_{i_{1}<i_{2}<\cdots<i_{t}} \operatorname{Pr}\left(E_{i_{1}} \cap \cdots \cap E_{i_{\ell}}\right)
\end{aligned}
$$