2008/08/06 by Matthew Macauley, Macauley, Matthew, Brian Rabern +3
Mathematics · #03C99 #03F03 #54E45 #FOS: Mathematics #History and Overview (math.HO) #Logic (math.LO) #Metric Geometry (math.MG) #math.HO #math.LO #math.MG #msc:03C99 #msc:03F03 #msc:54E45
paper · pdf · doi:10.48550/arxiv.0808.0844
arxiv created 2008/08/06 · arxiv updated 2009/12/01
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.