A 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 [4] proved that $(x_n)$ is almost convergent to $L$ 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.

Bibliography

1

G. Bennett and N.J. Kalton: Consistency theorems for almost convergence. Trans. Amer. Math. Soc., 198:23-43, 1974.

2

J. Boos: Classical and modern methods in summability. Oxford University Press, New York, 2000.

3

Jeff Connor and K.-G. Grosse-Erdmann: Sequential definitions of continuity for real functions. Rocky Mt. J. Math., 33(1):93-121, 2003.

4

G. G. Lorentz: A contribution to the theory of divergent sequences. Acta Math., 80:167-190, 1948.