A Characterization of Cesaro Convergence

We show that a real bounded sequence $(x_n)$ is Ces`aro convergent to $\mathrm{\ell }$ if and only if the sequence of averages with indices in $[{\alpha }^k,{\alpha }^{k+1})$ converges to $\mathrm{\ell }$ for all $\alpha >1$. If, in addition, the sequence $(x_n)$ is nonnegative, then it is Ces`aro convergent to $0$ if and only if the same condition holds for some $\alpha >1$.