# Final Question(For Now) On Basics Limit Arithmetics

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.

Thanks a lot.

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What have you tried so far? –  Aryabhata Apr 1 '12 at 19:42
This one is worth thinking about for a long time on your own before seeking hints. –  Antonio Vargas Apr 1 '12 at 19:50

## 1 Answer

It's false: take $a_n = 1$ if $n = 2^m$ for some $m$, and $0$ otherwise.

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Very nice!$\ \$ –  David Mitra Apr 1 '12 at 19:55