Let's consider the simplest situation. On the one dimensional line of integers, and we starts from the origin. Each time we either move left or right (at the same probability) for 1 unit. How do I show that, for however large $m$, the probability that we eventually go farther than $m$ units from the origin is $1$?
You might want to consider the following question: what's the probability of never going left (or right) more than $2m$ times in a row? What's the probability of always staying in the $[-m,m]$ interval in light of the previous answer?