A Characterization of Cesaro Convergence

Abstract

We show that a real bounded sequence $(x_n)$ is Ces\`aro convergent to $\ell$ if and only if the sequence of averages with indices in $[\alpha^k,\alpha^{k+1})$ converges to $\ell$ 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 $\alpha>1$.