= Solution
Let $A_n=\{X_m=0\text{ for some }m\geq n\}$. By the <Markov property>,
$$
v(X_n)=\mathbb P(A_n\mid\mathcal F_n).
$$
The events $A_n$ decrease to the event that the walk visits zero infinitely often, which has probability zero by the stated transience assumption. Hence
$$
\mathbb E[v(X_n)]=\mathbb P(A_n)\longrightarrow0.
$$
Part a and the <almost sure supermartingale convergence theorem> give an almost-sure limit $V\geq0$. By <Fatou lemma>, $\mathbb EV\leq\liminf_n\mathbb E[v(X_n)]=0$, so $V=0$ almost surely.
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