In Rick Durrett's book "Probability: Theory and Examples", there is a theorem about martingales with bounded increment:
Theorem 5.3.1. Let $X_1, X_2, \ldots$ be a martingale with $|X_{n+1} − X_n | \le M < \infty$. Let \begin{align} & C = \{\lim X_n \text{ exists and is finite}\} \\ & D = \{\limsup X_n = +\infty \text{ and } \liminf X_n = −\infty\} \end{align} Then $P (C \cup D) = 1$.
But later in the book (Exercise 5.3.2.), the author states that
... This example shows that it is not enough to have $\sup |X_{n+1} − X_n| < \infty$ in Theorem 5.3.1.
I'm confused of why these two conditions are not the same. If $\sup |X_{n+1} − X_n| < \infty$, shouldn't there exists an $M<\infty$ such that $|X_{n+1} − X_n| \le M$ for all $n$?
In the first condition $|X_{n+1}(\omega)-X_n(\omega)|\leqslant M$ for every $n$ and for almost every $\omega$, for some finite constant $M$. In the second condition $|X_{n+1}(\omega)-X_n(\omega)|\leqslant M(\omega)$ for every $n$ and every $\omega$, for some random variable $M$ almost surely finite.