Characterizing the Infinitely Visited Intersections in a Ride-Forever Path on a Directed Graph

Let $v, w \in RF(p)$. All we need to do is to show that there is a path from $v$ to $w$: since we chose an arbitrary pair of vertices, this means that any vertex of $RF(p)$ can be reached by any other vertex of $RF(p)$, thus $RF(p)$ is strongly connected.

Let $N \in \mathbb{N}$ arbitrary. Since $v \in RF(p)$, we visit $v$ an infinite amount of times, so there exists $n > N$ such that after $n$ steps we are on $v$. Since $w$ is also visited an infinite amount of times, there exists $m > n$ such that after $m$ steps we are on $v$. But since $m > n$, there must be a path from $v$ to $w$, otherwise we couldn't get to $w$ after visiting $v$. It follows that $RF(p)$ is strongly connected.