Question about the favorite sites of a simple random walk

Question

Let $F(n)$ be the set of favorite sites (most visited sites) of a simple random walk on $\mathbb{Z}$ at time $n$. I've read from multiple sources that $P(F(n)=1 \text{ i.o.})=1$ and $P(F(n)=2\text{ i.o.})=1$. While the first claim makes sense to me, since it's due to the recurrence of the SRW, I'm not quite sure why $P(F(n)=2\text{ i.o.})=1$ is necessarily true. I mean, I also happen to know that the maximum favorite site tends to infinity as $n\rightarrow \infty$ almost surely, based on a paper by Bass and Griffin (1985), so constantly-changing maximum favorite sites imply the claim. However, is there anything obvious about the simple random walk that make it clear that two simultaneous favorite sites occur infinitely often w.p. 1? It could be just a simple idea, but I just can't see it right now.