Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 2 / b / Solution

Let $A_n=\{X_m=0\text{ for some }m\ge n\}$. By the Markov property,The events $A_n$ decrease to the event that the walk visits zero infinitely often, which has probability zero by the stated transience assumption. HencePart a and the almost sure supermartingale convergence theorem give an almost-sure limit $V\ge 0$. By Fatou lemma, $\mathbb{E}V\le {\mathrm{liminf}}_n\mathbb{E}[v(X_n)]=0$, so $V=0$ almost surely.