A Novel Proof of the HeineBorel Theorem
Abstract
Every beginning real analysis student learns the classic HeineBorel 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 HeineBorel theorem is an immediate corollary.
 Publication:

arXiv eprints
 Pub Date:
 August 2008
 arXiv:
 arXiv:0808.0844
 Bibcode:
 2008arXiv0808.0844M
 Keywords:

 Mathematics  History and Overview;
 Mathematics  Logic;
 Mathematics  Metric Geometry;
 54E45;
 03C99;
 03F03