Probabilities commute with limits taken with respect to non-decreasing and decreasing sets of events.

Proof for non-decreasing sets

Let ${C_{n}}_{n \in N}$ be a non-decreasing sequence, i.e. $C_{n} \subseteq C_{n + 1}$ , “converging” to $n \in N ⋃ C_{n}$ . Then we seek to prove

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

For that, note that this sequence “grows radially” from $C_{1}$ . For that, define “rings” to be

{R_{1} \equiv C_{1} R_{n + 1} \equiv C_{n + 1} \cap C_{n}^{c}

With that, we see that $⋃ C_{n} = ⋃ R_{n}$ ¹.
Note also that $R_{n} \cap R_{n + 1} = \emptyset$ , and, thus, are all mutually exclusive.
Note that, since $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ , we have that

P (R_{n + 1}) = P (C_{n + 1} \cap C_{n}^{c}) = P (C_{n + 1}) + (1 - P (C_{n}^{c})) - = 1 P (C_{n + 1} \cup C_{n}^{c}) = P (C_{n + 1}) - P (C_{n})

Given all that, note that

P (n \to \infty lim C_{n}) = P (n \in N ⋃ C_{n}) = P (n \in N ⋃ R_{n}) = (2) n \in N \sum P (R_{n}) = (3) n \to \infty lim [P (C_{1}) + n = 2 \sum n (P (C_{n + 1}) - P (C_{n}))] = n \to \infty lim P (C_{n})

Proof for decreasing sets

Let ${D_{n}}_{n \in N}$ be a non-decreasing sequence, i.e. $D_{n} \supseteq D_{n + 1}$ , “converging” to $n \in N ⋂ D_{n}$ . Then we seek to prove

n \to \infty lim P (D_{n}) = P (n \to \infty lim D_{n}) = P (n \in N ⋂ D_{n})

Note that $D_{n} \supseteq D_{n + 1} ⟹ D_{n}^{c} \subseteq D_{n + 1}^{c}$ , we have

P (n \to \infty lim D_{n}) = P (n \in N ⋂ D_{n}) = 1 - P (n \in N ⋃ D_{n}^{c}) = 1 - n \to \infty lim P (D_{n}^{c}) = n \to \infty lim (1 - P (D_{n}^{c})) = n \to \infty lim P (D_{n})