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