= Solution
For <simple random walk> on $\mathbb Z^d$, $d\geq3$, the <Green-function decay for simple random walk on the integer lattice> and the <Strong Markov property> give
$$
P_0(H_{\{x_k\}}<\infty)
=\frac{g(0,x_k)}{g(x_k,x_k)}
\leq C|x_k|^{2-d}
=C2^{-k(d-2)}.
$$
The series over $k$ converges, so the first of the <Borel-Cantelli lemmas> says that almost surely only finitely many of the points $x_k$ are ever hit. Moreover, $\mathbb Z^d$ is a <transient graph> for $d\geq3$, so each of those finitely many points is visited only finitely often. Therefore the <simple random walk> visits $A$ only finitely often almost surely:
$$
P_0(X_n\in A\text{ for infinitely many }n)=0.
$$
Solved by gpt-5.6-sol high.
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