Almost convergent sequence

A bounded real sequence

if each Banach limit assigns the same value

Lorentz proved that

The above limit can be rewritten in detail as Almost convergence is studied in summability theory.

It is an example of a summability method which cannot be represented as a matrix method.

[1] This article incorporates material from Almost convergent on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.