# Tagged: decreasing sequence of events

## Problem 744

A sequence of events $\{E_n\}_{n \geq 1}$ is said to be increasing if it satisfies the ascending condition
$E_1 \subset E_2 \subset \cdots \subset E_n \subset \cdots.$ Also, a sequence $\{E_n\}_{n \geq 1}$ is called decreasing if it satisfies the descending condition
$E_1 \supset E_2 \supset \cdots \supset E_n \supset \cdots.$

When $\{E_n\}_{n \geq 1}$ is an increasing sequence, we define a new event denoted by $\lim_{n \to \infty} E_n$ by
$\lim_{n \to \infty} E_n := \bigcup_{n=1}^{\infty} E_n.$

Also, when $\{E_n\}_{n \geq 1}$ is a decreasing sequence, we define a new event denoted by $\lim_{n \to \infty} E_n$ by
$\lim_{n \to \infty} E_n := \bigcap_{n=1}^{\infty} E_n.$

(1) Suppose that $\{E_n\}_{n \geq 1}$ is an increasing sequence of events. Then prove the equality of probabilities
$\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right).$ Hence, the limit and the probability are interchangeable.

(2) Suppose that $\{E_n\}_{n \geq 1}$ is a decreasing sequence of events. Then prove the equality of probabilities
$\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right).$