It seems to me a sequence that is [almost convergent](https://en.wikipedia.org/wiki/Almost_convergent_sequence) implies that its [Cesàro means](http://en.wikipedia.org/wiki/Ces%C3%A0ro_mean) converges but not vice versa.

What is the example that a not almost convergent sequence whose Cesàro means converge. i.e. for a sequence $\{x\_i\}\_{k \in \mathbb N}$

> 
> There exists an $L$, for all $\epsilon$, for all $n$, there exists a $p\_0(n)$, such that for all $p > p\_0(n)$, we have
>  $$\left\lvert \frac{\sum\_{k=1}^{p} x\_{n+k-1}}{p}-L \right\rvert < \epsilon$$
>  and that
>  $\lim\_{n \to \infty}p\_0(n) = +\infty$
> 
> 
>