# Limit of sequence differences

1. Oct 24, 2012

### Jakim

We know that for $k\in\mathbb{N}$ we have, if:

$\displaystyle\lim_{n\to\infty}\left(a_{n}-a_{n-k}\right)=k\cdot a$

then:

$\displaystyle\lim_{n\to\infty}\frac{a_n}{n}=a$

When the reverse impliaction is also true? What do we have to assume to achieve if and only if theorem? I'm especially interested in a case when $a=0$. I state (now without a proof, just intuition) that if

$\displaystyle\lim_{n\to\infty}\frac{a_n}{n}=a$

then $\displaystyle\limsup_{n\to\infty}\left(a_{n}-a_{n-k}\right)$ and $\displaystyle\liminf_{n\to\infty}\left(a_{n}-a_{n-k}\right)$ are bounded and if these limits are the same, they are equal $0$.

Last edited: Oct 24, 2012