Five Circles: Real Analysis Theorems equivalent to Completeness
Rafael Cantuba
TL;DR
This work develops a topology-centered exposition of five interlocking circles of real-analysis theorems, each circle being a finite chain of implications that are all equivalent to Dedekind completeness of the reals. It builds a unified framework using filter-based limits, supremum concepts, compactness arguments, and elementary integration to show how convergence, connectedness, differentiation, compactness, and integration are intimately tied to completeness. By detailing preliminaries and then presenting each circle with its constituent theorems and their interdependencies, the paper clarifies how many core results of real analysis ultimately depend on the same completeness axiom. The approach highlights the role of topological ideas in proving classical calculus results and provides an accessible entry point to the network of equivalences for advanced undergraduates and beginning graduates. The synthesis underscores the practical impact of completeness: a broad suite of theorems can be viewed as manifestations of the same foundational property of $\oldsymbol{\mathbb{R}}$.
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.