I have the following problem on random walks:
Consider a $d$-dimensional symmetric random walk which starts at the origin at time $n=0$. Show that the walk has probability $1$ of returning infinitely often to a position already previously occupied.
This problem doesn't make much sense to me because we know that symmetric random walks are transient for $d\geq 3$. Since the random walk starts at the origin, we would show that the random walk returns to the origin infinitely often. Doesn't this contradict the fact that the random walk is transient, or am I missing something?