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

Lorentz proved that $(x\_n)$ is almost convergent if and only if

- $limlimits\_\{ptoinfty\}\; frac\{x\_\{n\}+ldots+x\_\{n+p-1\}\}p=L$

The above limit can be rewritten in detail as

- $(forall\; varepsilon>0)\; (exists\; p\_0)\; (forall\; p>p\_0)\; (forall\; n)\; left|frac\{x\_\{n\}+ldots+x\_\{n+p-1\}\}p-Lright|math>$

- 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.

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Last updated on Tuesday August 26, 2008 at 11:50:37 PDT (GMT -0700)

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Last updated on Tuesday August 26, 2008 at 11:50:37 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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