Almost convergent sequence

A bounded real sequence ${\displaystyle (x_{n})}$ is said to be almost convergent to ${\displaystyle L}$ if each Banach limit assigns the same value ${\displaystyle L}$ to the sequence ${\displaystyle (x_{n})}$.

Lorentz proved that ${\displaystyle (x_{n})}$ is almost convergent if and only if

${\displaystyle \lim \limits _{p\to \infty }{\frac {x_{n}+\ldots +x_{n+p-1}}{p}}=L}$

uniformly in ${\displaystyle n}$.

The above limit can be rewritten in detail as

${\displaystyle \forall \varepsilon >0:\exists p_{0}:\forall p>p_{0}:\forall n:\left|{\frac {x_{n}+\ldots +x_{n+p-1}}{p}}-L\right|<\varepsilon .}$

Almost convergence is studied in summability theory. It is an example of a summability method which cannot be represented as a matrix method.^[1]