Suppose we have that $X_i$ are iid random variables with $P(X_i =1) = P(X_i = -1) = 1/2$ and that $X_0 = 0$. Then, we define the simple symmetric random walk to be $S_n = \sum_{i=1}^n X_i$. We define the hitting time as $\tau_{i,j} = min\{n \geq 1 : S_n = j \ |\ S_0 = i\}$. I would like to show that $P(\tau_{0,0} < \infty) = 1$.
Specifically, I would like to use the Borel-Cantelli lemma to show that if I can define some independent sequence of events here, and show that the sum of the events is infinite, then moves of some finite length happen infinitely often, and so I can show that it must return to $0$. However, I am not sure how to define such an indexed event. Could someone give me a hint? Thank you