## Interchangeability of Limits and Probability of Increasing or 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

\[\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right). \]