Non-Compactness of a Closed and Bounded Set

Abstract

If every closed and bounded set in a metric space is compact, the space is said to have the Heine-Borel property. This property holds in every finite dimensional normed space, but may not be true in general. Though its proof appears in many basic analysis courses, it is hard to motivate as the result is subtle and the applications are not obvious. Our goal is to provide an elegant proof as a resource for teachers that will enable them to motivate the study of this essential property and to understand the mathematics in it as a valuable teaching tool.