Showing that a_n > 0 given its decreasing and lim a_n=0

  • #1
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I'm reading a proof, and it says that since ##\lim_{n\to\infty}a_n = 0## and ##a_{n+1}\le a_n## for all ##n##, it must be the case that ##a_n \ge 0##.

This seems very obvious, since if ##a_n < 0## for some ##n## then it would decreasing from then on, so there wouldn't be infinitely many elements contained in every ##\epsilon##-neighborhood around ##0##. But is this sufficient for proof? What would a write up of a "proof" of this simple statement look like?
 

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  • #2
andrewkirk
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Assume the opposite. Then there's some ##a_n<0##. Then show that every subsequent member of the series is at least distance ##|a_n|## from 0. So 0 can't be the limit.
 

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