= Solution
{c}
Let $E_n=\{|X_n-1|=n\}$. These events are independent and $\sum_{n\geq2}\mathbb P(E_n)=\sum_{n\geq2}1/(n\log n)=\infty$. By the <Borel-Cantelli second lemma>, they occur infinitely often almost surely.
If $S_n/n$ had any finite sample-path limit, then
$$
\frac{X_{n+1}}n=\frac{S_n}n-\frac{n-1}{n}\frac{S_{n-1}}{n-1}\longrightarrow0.
$$
But on $E_{n+1}$, $|X_{n+1}|/n$ is either $1$ or $(n+2)/n$, contradicting this convergence infinitely often. Therefore \b[the averages have no finite almost sure limit]. This is a <rare-jump counterexample to a strong law with convergent first-moment averages>.
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