Prove or Disprove: if $\lim\limits_{n \to \infty} (a_{2n}-a_n)=0,$ then $\lim\limits_{n \to \infty} a_n=0.$ Prove or Disprove: if $\lim\limits_{n \to \infty} (a_{2n}-a_n)=0,$ then $\lim\limits_{n \to \infty} a_n=0.$
I don't think that this is true. and I'm trying to think about counterexample, but couldn't figure out a mathematical form of the sequence that I thought about, its a sequence where certain terms that are multipliers of $2n$ and $n$ become more close to each other as $n$ grows.
any hints ?  

Just let $a_n = 1$ for all $n$, for example.
