# 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 ?

• What about the constant sequence $a_n=1$? – Olivier Bégassat Nov 16 '15 at 10:23
• Maybe also interesting: Does $a_n$ converge if $a_{2n}-a_n$ converges? In the moment of typing I don't know the answer. – fweth Nov 16 '15 at 11:08
• @fweth Let $a_n = \gcd(3,n)$. Then $a_{2n} = a_n$ for all $n$, but $a_n$ does not converge. – Najib Idrissi Nov 16 '15 at 12:34

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