A new construction of the real numbers by alternating series

TL;DR

This paper introduces a novel method for constructing the real numbers from rationals using alternating series, generalizing previous approaches and demonstrating multiple construction methods with applications to irrationality proofs.

Contribution

It presents a new construction technique for real numbers that extends existing methods and shows the existence of infinitely many such constructions.

Findings

Multiple constructions of real numbers from rationals are possible.

The method generalizes previous constructions by Knopfmacher and Knopfmacher.

Application to proving the irrationality of specific numbers.

Abstract

We put forward a new method of constructing the complete ordered field of real numbers from the ordered field of rational numbers. Our method is a generalization of that of A. Knopfmacher and J. Knopfmacher. Our result implies that there exist infinitely many ways of constructing the complete ordered field of real numbers. As an application of our results, we prove the irrationality of certain numbers.