# What's the easiest way to show that a random walk can go arbitrarily far?

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?
@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