# If $a_{2n}-a_n\to 0$, does the sequence $a_n$ have a limit?

Is this claim true or false?

Given $\lim \limits_{n\to \infty}\ (a_{2n}-a_n)=0$ then $\lim \limits_{n\to \infty}\ a_n$ exists.

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This one is worth thinking about for a long time on your own before seeking hints. – Antonio Vargas Apr 1 '12 at 19:50
False. Take $a_n=\log\log n$ – A.S. Oct 9 '15 at 22:04

It's false: take $a_n = 1$ if $n = 2^m$ for some $m$, and $0$ otherwise.
Very nice!$\ \$ – David Mitra Apr 1 '12 at 19:55