= Solution
{c}
For every $m$, $D_m$ is a continuous square-integrable vector <martingale>. Its norm is a <submartingale>, so the <Doob L2 maximal inequality> and the estimate from part (c) give
$$
\mathbb E\sup_{t\leq T}|D_m(t)|^2\leq4|a|^2\frac{(KT)^{m+1}}{(m+1)!}.
$$
The <Cauchy-Schwarz inequality> now shows
$$
\sum_m\mathbb E\sup_{t\leq T}|D_m(t)|
\leq2|a|\sum_m\frac{(KT)^{(m+1)/2}}{\sqrt{(m+1)!}}<\infty.
$$
By <Tonelli's theorem>, the random sum of these suprema is finite almost surely. Therefore the telescoping series $a+\sum_mD_m(t)$ converges absolutely and uniformly on $[0,T]$ outside one null event. Its limit is continuous and agrees at each fixed time with the $L^2$ limit from part (c). Taking the intersection of the probability-one events for integer $T$ provides a single version on the whole half-line. Hence
$$
\boxed{\sup_{t\leq T}|y_t^{(n)}-y_t^{(\infty)}|\longrightarrow0\quad\text{almost surely for every finite }T.}
$$
This also gives the requested pointwise almost-sure convergence of the full sequence.
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