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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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up vote 5 down vote accepted

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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@Voldemort: I don't think it is prudent to calculate it exactly. Just look at infinite series of coin tosses and ask yourself what's the probability that in each interval $(m(2n+1)+1,\ldots ,(m+1)(2n+1))$ we have at least one tails, by multiplying the probability for each interval separately. –  tomasz Sep 24 '12 at 1:54
    
I see it. It's at least the probability that an arbitrary $2m$ of heads/tails show in a roll, which is eventually 1. –  Voldemort Sep 24 '12 at 1:56
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