Almost convergent sequence

A bounded real sequence $(x\_n)$ is said to be "almost convergent" to $L$ if each Banach limit assignsthe same value $L$ to the sequence $(x\_n)$.

Lorentz proved that $(x\_n)$ is almost convergent if and only if:$limlimits\_\{p\; oinfty\}\; frac\{x\_\{n\}+ldots+x\_\{n+p-1p=L$uniformly in $n$.

The above limit can be rewritten in detail as:

References

* G. Bennett and N.J. Kalton: "Consistency theorems for almost convergence." Trans. Amer. Math. Soc., 198:23--43, 1974.

* J. Boos: "Classical and modern methods in summability." Oxford University Press, New York, 2000.

* J. Connor and K.-G. Grosse-Erdmann: "Sequential definitions of continuity for real functions." Rocky Mt. J. Math., 33(1):93--121, 2003.

* G.G. Lorentz: "A contribution to the theory of divergent sequences." Acta Math., 80:167--190, 1948.