# True or false: if $(a_{n+1} -a_n)$ is a bounded sequence, $\lim\limits_{n\mapsto \infty}(\frac{a_n}{n^2})=0$

Need help with this question:

True or false: if $(a_{n+1} -a_n)$ is a bounded sequence, then $\lim\limits_{n\mapsto \infty}(\frac{a_n}{n^2})=0$

I couldn't find counterexample and couldn't not prove it.

• True by Cesàro-Stolz or by telescoping, of course. The exponent $2$ can be replaced by any exponent $>1$ and the claim still is true. – Jack D'Aurizio Nov 11 '17 at 14:04

$$a_n = a_0 + \sum_{k=1}^n (a_k - a_{k-1}).$$
• $a_0$ is not defined as $n$ is a natural number – Avishay28 Nov 11 '17 at 13:18
• Note that you can use $a_1$ and start the sum with $k=2$. – ncmathsadist Nov 11 '17 at 13:19