I'm trying to understand a pretty standard proof about the possible events for Martingales with bounded increments. Specifically, assume
1) $X_1, X_2, \ldots$ is a martingale
2) $|X_{n+1} - X_n| \le M$
3) $C = \{ \omega : \lim X_{n}(\omega) \text{ exists and is finite} \}$
4) $D = \{ \omega : \lim\sup X_{n}(\omega) = \infty, \lim \inf X_n(\omega) = - \infty \}$
Then $P(C \cup D) = 1$
In particular, the proof in Durrett, page 204-205, they mention that if I fix a constant $K$ and let $N$ be the first time $X_n \le -K$, then I can use the fact that a non-negative (super)martingale has a limit almost surely on $ \{ \omega : N(\omega) = \infty \}$. It's precisely this last statement, "on ${ \omega : N(\omega) = \infty }$" that I don't understand. Can someone clarify why this doesn't apply for $ \{ \omega : N(\omega) < \infty \}$?
Thanks!