= Solution
{c}
Set $A_n=\{\sup_{s\leq t}|M_s^{(n)}|\geq2^{-(n-1)/2}\}$. Part (a) gives $\sum_n\mathbb P(A_n)<\infty$. The <Borel-Cantelli first lemma> implies that, almost surely, only finitely many $A_n$ occur. On the complementary full-probability event, all sufficiently late terms satisfy
$$
\sup_{s\leq t}|M_s^{(n)}|<2^{-(n-1)/2}.
$$
These deterministic bounds form a convergent geometric series. The finitely many earlier suprema are finite by continuity on the compact time interval. Hence
$$
\boxed{\sum_n\sup_{s\leq t}|M_s^{(n)}|<\infty\quad\text{almost surely}.}
$$
The series therefore converges absolutely and uniformly on $[0,t]$ by the <Weierstrass M-test>, and its sample-path limit is continuous.
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