Solution
= Solution
Let $Z$ be a fair Bernoulli variable measurable at time zero and let $(S_n)$ be an independent simple symmetric random walk. Then $X_n=ZS_n$ is a martingale with increments bounded by one. On $\{Z=0\}$ it converges to zero, while on $\{Z=1\}$ the recurrence of the <simple symmetric random walk> gives limsup $+\infty$ and liminf $-\infty$. Thus $\mathbb P(A)=\mathbb P(B)=1/2$.