Increasing and Decreasing Sequence of Events

Definition

$\underset{n\to \infty }{\lim }{C}_n=\{\begin{array}{ll}\bigcup _{n=1}^{\infty }{C}_n& \text{for increasing sequence}\\ \bigcap _{n=1}^{\infty }{C}_n& \text{for decreasing sequence}\end{array}$

Facts

Let $\{C_n\}$ be an increasing sequence of events, then $\underset{n\to \infty }{\lim }P(C_n)=P(\underset{n\to \infty }{\lim }{C}_n)=P(\bigcup _{n=1}^{\infty }{C}_n)$

Let $\{C_n\}$ be a decreasing sequence of events, then $\underset{n\to \infty }{\lim }P(C_n)=P(\underset{n\to \infty }{\lim }{C}_n)=P(\bigcap _{n=1}^{\infty }{C}_n)$