TITLE: Martingale with bounded increment.
QUESTION [6 upvotes]: 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 $n$?

REPLY [8 votes]: 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.