On flexible sequences

TL;DR

This paper introduces the concept of flexible sequences within nonstandard analysis, where sequence terms are external numbers with algebraic properties similar to real numbers, and explores their convergence behaviors.

Contribution

It defines flexible sequences and their convergence notions, extending classical sequence properties to this new framework within nonstandard analysis.

Findings

Flexible sequences are stable under shifts, additions, and multiplications.

Two forms of convergence are introduced and related.

Classical convergence properties are adapted to flexible sequences.

Abstract

In the setting of nonstandard analysis we introduce the notion of flexible sequence. The terms of flexible sequences are external numbers. These are a sort of analogue for the classical \emph{O$ (\cdot ) $} and \emph{o$ (\cdot ) $} notation for functions, and have algebraic properties similar to those of real numbers. The flexibility originates from the fact that external numbers are stable under some shifts, additions and multiplications. We introduce two forms of convergence, and study their relation. We show that the usual properties of convergence of sequences hold or can be adapted to these new notions of convergence and give some applications.