Bonferroni's Inequality

Let $P(E_i)$ be the probability that $E_i$ is true, and $P\left({\bigcup_{i=1}^n E_i}\right)$ be the probability that $E_1$, $E_2$, ..., $E_n$ are all true. Then

$$P\left({\,\bigcup_{i=1}^n E_i}\right)\leq \sum_{i=1}^n P(E_i).$$