theorem-limit-theorem-for-monotone-events

Theorem

If ${C_{n}}$ is increasing

Then

n \to \infty lim P (C_{n}) = P (n \to \infty lim C_{n}) = P (n = 1 ⋃ \infty C_{n})

If ${C_{n}}$ is decreasing

Then

n \to \infty lim P (C_{n}) = P (n \to \infty lim C_{n}) = P (n = 1 ⋂ \infty C_{n})

Let $R_{1} = C_{1}$ be any set and let $R_{n} = C_{n} \cap C_{n - 1}^{C}$ for $n > 1$

It follows that $⋃_{n = 1}^{\infty} C_{n} = ⋃_{n = 1}^{\infty} R_{n}$ and that $R_{m} \cap R_{n} = ϕ$ , for $m \neq = n$ .

Also, $P (R_{n}) = P (C_{n}) - P (C_{n - 1})$ .

Applying the third axiom of probability set function yields the following string of equalities:

P [n \to \infty lim C_{n}] = P (n = 1 ⋃ \infty C_{n}) = P (n = 1 ⋃ \infty R_{n}) = n = 1 \sum \infty P (R_{n}) = n \to \infty lim j = 1 \sum n P (R_{j}) = n \to \infty lim {P (C_{1}) + j = 2 \sum n [P (C_{j}) - P (C_{j - 1})]} = n \to \infty lim P (C_{n})