This is fairly standard. Let Xi be a symmetric random walk and let Ei be the event that Xi = 0. Then and . The diagram shows the number of ways of getting to a particular node. The dashed arrows are meant to represent that the random walk then continues.
In general, at time 2n, the number of ways of getting to 0 is . Thus
Stirling’s approximation tells us that
Thus we have:
By the second Borel-Cantelli Lemma, since the Ei are independent,
Thus the symmetric random walk returns to the origin infinitely often.