If a dot is moving (from zero) left or right, by one, with 50% chance to go left or right - is it going to go to the +inf or -inf when it has infinite moves?
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This is the simple symmetric random walk on the integers. It is well known that this walk is recurrent, and so visits every point infinitely often with probability one. In particular, it has probability zero of converging to either $+\infty$ or $-\infty$. Proofs can be found here, or in most introductory textbooks on random processes.