Almost convergent sequence

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 ) .

Lorentz proved that ( x n ) is almost convergent if and only if

uniformly in n .

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.