= Solution
{c}
Apply part (i) for every positive integer $K$ and discard the <countable union> of its exceptional <null sets>. A <sample path> bounded below never reaches $-K$ for some such $K$, and must therefore converge to a finite <limit of a sequence>.
Apply the same argument to the <martingale> $(-M_n)$. Outside one more <null set>, every <sample path> bounded above also converges finitely. Thus a <sample path> which does not converge finitely must be unbounded in both directions. Each finite initial segment has finite values, so removing any such segment preserves unboundedness. It follows that the <limit superior> is $+\infty$ and the <limit inferior> is $-\infty$. Consequently
$$
\boxed{\mathbb P(A\cup B)=1.}
$$
This proves the <bounded-increment martingale convergence-or-oscillation dichotomy>; in particular one-sided divergence to infinity is impossible with positive <probability>.
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