theorem-boole's-inequality

Theorem

Let ${C_{n}}$ : Any sequence of events

Then

P (n = 1 ⋃ \infty C_{n}) \leq n = 1 \sum \infty P (C_{n})

Let $D_{n} = ⋃_{i = 1}^{n} C_{i}$ . Then ${D_{n}}$ is an increasing sequence of events that go up to $⋃_{n = 1}^{\infty} C_{n}$ .

Also, for all $j$ , $D_{j} = D_{j - 1} \cup C_{j}$ . Hence, by point 5 of Property of Probability Set Function,

P (D_{j}) \leq P (D_{j - 1}) + P (C_{j})

that is,

P (D_{j}) - P (D_{j - 1}) \leq P (C_{j})

In this case, the $C_{i}$ s are replaced by the $D_{i}$ s in expression