2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/168873Every 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.History and OverviewLogicMetric Geometry54E45; 03C99; 03F03text