A Novel Proof of the Heine-Borel Theorem
History and Overview
2008-09-12 v1 Logic
Metric Geometry
Abstract
Every beginning real analysis student learns the classic Heine-Borel theorem, that the interval [0,1] is compact. In this article, we present a proof of this result that doesn't involve the standard techniques such as constructing a sequence and appealing to the completeness of the reals. We put a metric on the space of infinite binary sequences and prove that compactness of this space follows from a simple combinatorial lemma. The Heine-Borel theorem is an immediate corollary.
Keywords
Cite
@article{arxiv.0808.0844,
title = {A Novel Proof of the Heine-Borel Theorem},
author = {Matthew Macauley and Brian Rabern and Landon Rabern},
journal= {arXiv preprint arXiv:0808.0844},
year = {2008}
}