I was asked to prove the said equivalence for a simple random walk on the integers (which may be biased): $X_{i}\in\left\{ \pm1\right\} $, $S_{n}=S_{0}+\sum_{i=1}^{n}X_{i}$.
The first direction is pretty direct: assume that $\sum_{i=0}^{\infty}\mathbb{P}\left(S_{i}=S_{0}\right)<\infty$, then according to the Borel–Cantelli lemma, we have $\mathbb{P}\left(S_{i}=S_{0}\quad i.o.\right)=0$.
I am having trouble with the other direction. Ultimately, I would like to use the counterpart of B-C, but $S_i=S_0$ and $S_j=S_0$ might be dependnet (in the case $\mathbb{P}\left(X_{i}=\pm1\right)\neq0.5$). If we define $Z=\#\left\{ i\mid S_{i}=S_{0}\right\} $ we know that $\mathbb{E}\left[Z\right]=\sum_{i=0}^{n}\mathbb{P}\left(S_{i}=S_{0}\right)=\infty$, but I don't know how to proceed from here. I have seen this question which seems very relevant, but it is rather not detailed.