Past exam of the probability course of the University of Oxford / 2023 / 1 / c / ii / Solution

If $0<b\le 1$, then $0<U_j\le 1$ almost surely. Intersecting these events over all $j$ still gives an event of probability one. On it, $0\le W_{n+1}\le W_n$, so the products have a finite nonnegative limit $W_{\infty }$.

For every $\epsilon >0$, the inclusion $\{W_{\infty }>\epsilon \}\subseteq \{W_n>\epsilon \}$ and part (c)(i) imply $\mathbb{P}(W_{\infty }>\epsilon )=0$. Taking $\epsilon =1/m$ for positive integers $m$ proves $W_{\infty }=0$ almost surely. ThusThe monotonicity is the extra ingredient that upgrades convergence in probability to almost sure convergence here.