On distribution and almost convergence of bounded sequences

TL;DR

This paper introduces new concepts of properly and simply distributed sequences, proves their almost convergence, and generalizes previous work on Banach limits, advancing the understanding of bounded sequence behavior.

Contribution

It presents a generalized theory of distribution and almost convergence, extending prior results by Feng and Li to broader classes of sequences.

Findings

Properly and simply distributed sequences are almost convergent.

The generalized theory encompasses previous results on Banach limits.

Provides new insights into the behavior of bounded sequences.

Abstract

In this paper, we give the concepts of properly distributed and simply distributed sequences, and prove that they are almost convergent. Basing on these, we review the work of Feng and Li [Feng, B. Q. and Li, J. L., Some estimations of Banach limits, J. Math. Anal. Appl. 323(2006) No. 1, 481-496. MR2262220 46B45 (46A45).], which is shown to be a special case of our generalized theory.