= Solution
Let independent <Bernoulli random variables> $Y_n$ have success probabilities $2^{-n}$, for $n\geq1$, and use their <natural filtration>. Define
$$
\boxed{X_0=0,\qquad X_n=\sum_{k=1}^n(1-2^kY_k).}
$$
Each increment is integrable, finite, and independent of the previous <filtration>, with <expectation>
$$
\mathbb E(1-2^nY_n)=1-2^n2^{-n}=0.
$$
Thus $X$ is a <martingale>. The increments have no common finite bound, since a success produces a jump of size $2^n-1$.
Since $\sum_n\mathbb P(Y_n=1)<\infty$, the <First Borel-Cantelli lemma> says that only finitely many successes occur <almost surely>. Consequently
$$
X_n=n-\sum_{k\geq1}2^kY_k
$$
for all sufficiently large $n$, with the random sum finite <almost surely>. Therefore \b[$X_n\to+\infty$ <almost surely>], so neither finite convergence nor two-sided oscillation occurs. This <rare-jump martingale divergence> example gives probability zero, rather than one, for the union in (i).
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