= Solution
If $0<b\leq1$, then $0<U_j\leq1$ <almost surely>. Intersecting these events over all $j$ still gives an event of <probability> one. On it, $0\leq W_{n+1}\leq W_n$, so the products have a finite nonnegative limit $W_\infty$.
For every $\varepsilon>0$, the inclusion $\{W_\infty>\varepsilon\}\subseteq\{W_n>\varepsilon\}$ and part (c)(i) imply $\mathbb P(W_\infty>\varepsilon)=0$. Taking $\varepsilon=1/m$ for positive integers $m$ proves $W_\infty=0$ <almost surely>. Thus
$$
\boxed{W_n\longrightarrow0\quad\text{almost surely for }0<b\leq1.}
$$
The monotonicity is the extra ingredient that upgrades <convergence in probability> to <almost sure convergence> here.
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