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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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    False. Take $a_n=\log\log n$2015-10-09

1 Answers 1

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

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    Very nice!$\ \ $2012-04-01