• Corpus ID: 16318865

A Novel Proof of the Heine-Borel Theorem

@article{Macauley2008ANP,
  title={A Novel Proof of the Heine-Borel Theorem},
  author={Matthew Macauley and Brian Rabern and Landon Rabern},
  journal={arXiv: History and Overview},
  year={2008}
}
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. 1. The Heine… 
The Nonstraddling Lemma and a new look at the Heine-Borel Theorem
Any family of intervals in the real line determines a linking equivalence relation on its union. The equivalence classes are order convex so are therefore themselves intervals, partitioning the union

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This Christmas Eve, many years ago, a group of children travelled through cold, dark forests where lean wolves roamed and white tailed rabbits hid from their train as it thundered through the quiet wilderness.
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