The Bonferroni inequality states that [Pleft(cap_{i} C_{i} ight) geq 1-sum_{i} Pleft(overline{C_{i}} ight)] (a) Show that this holds

The Bonferroni inequality states that

\[P\left(\cap_{i} C_{i}\right) \geq 1-\sum_{i} P\left(\overline{C_{i}}\right)\]

(a) Show that this holds for 3 events.

(b) Let \(C_{i}\) be the event that the \(i\) th confidence interval will cover the true value of the parameter for \(i=1, \ldots, m\). If \(P\left(\overline{C_{i}}\right) \leq \alpha / m\), so the probability of not covering the \(i\) th parameter is at most \(\alpha / m\), show that the probability that all of the confidence intervals cover their respective parameters is at least \(1-\alpha\).