Five Circles: Real Analysis Theorems equivalent to Completeness

TL;DR

This paper explores five interconnected theorems in real analysis, each equivalent to the Dedekind completeness of the real numbers, covering key concepts like convergence, connectedness, differentiability, compactness, and integration.

Contribution

It systematically presents the equivalences between these five fundamental theorems and the completeness property, clarifying their interrelations.

Findings

Each of the five theorems is shown to be equivalent to Dedekind completeness.

The work unifies key concepts in real analysis through these equivalences.

Provides a comprehensive exposition of Riemenschneider's framework.

Abstract

This is an exposition of the work of O. Riemenschneider about five ''circles'' of implications relating real analysis theorems each equivalent to the Dedekind completeness of the real field. These circles cover five elements of real function theory: convergence, connectedness, differentiability, compactness and integration.