Final Question(For Now) On Basics Limit Arithmetics

Question

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.

Answer 1

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