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
 * $$\lim\limits_{p\to\infty} \frac{x_{n}+\ldots+x_{n+p-1}}p=L$$

uniformly in $$n$$.

The above limit can be rewritten in detail as
 * $$\forall \varepsilon>0 : \exists p_0 : \forall p>p_0 : \forall n : \left|\frac{x_{n}+\ldots+x_{n+p-1}}p-L\right|<\varepsilon.$$

Almost convergence is studied in summability theory. It is an example of a summability method which cannot be represented as a matrix method.