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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 $1$?

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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?

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    I see it. It's at least the probability that an arbitrary $2m$ of heads/tails show in a roll, which is eventually 1.2012-09-24