A Novel Proof of the Heine-Borel Theorem

TL;DR

This paper introduces a new proof of the Heine-Borel theorem by applying combinatorial methods to the space of infinite binary sequences, avoiding traditional real analysis techniques.

Contribution

It provides a novel combinatorial proof of the Heine-Borel theorem using metric space concepts on binary sequences, differing from standard approaches.

Findings

Proves the compactness of the space of infinite binary sequences.

Derives the Heine-Borel theorem as a corollary from the combinatorial lemma.

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.