Theorem: Let $(X_n,\mathcal{F}_n)_{n\geq 0}$ be a martingale with $|X_n - X_{n-1}| \leq M<\infty$. Let $C := \{{\lim X_n \ \mbox{exists and} < \infty}\} $ and $D:=\{\limsup X_n = +\infty \mbox{and} \liminf X_n = -\infty\}$. Then $\mathbb{P}(C\cup D)=1$. (Reference: Durrett, Probability: theory and examples)
My question: Is it possible to get $\mathbb{P}(C\cup D)=0$, if I omit the condition $|X_n - X_{n-1}| \leq M<\infty$?
