Canonical Cauchy sequences for real numbers

TL;DR

This paper introduces a canonical method to represent real numbers using infinite integer sequences derived from continued fractions with subtractions, establishing a unique and ordered correspondence between sequences and real numbers.

Contribution

It provides a novel canonical construction of real numbers via continued fractions with subtractions, linking sequences to real numbers with a clear order-preserving correspondence.

Findings

Each sequence yields a unique decreasing Cauchy sequence of rationals.

The correspondence preserves the standard order of real numbers.

The method characterizes real numbers through infinite integer sequences.

Abstract

Based on continued fractions with subtractions, we identify the set of real numbers with the set of infinite integer sequences with all terms but the first one greater or equal to two. Each such sequence produces in a canonical way a unique strictly decreasing Cauchy sequence of rationals which converges to the corresponding real number. The correspondence is such that the standard order of real numbers is translated to the lexicographic order of sequences.