A Note on Using Unbounded Functions on Totally Bounded Sets in the Context of the Lebesgue Covering Lemma

Abstract

From a real-valued function , unbounded on a totally bounded subset of a metric space, we construct a Cauchy sequence in on which is unbounded. Taking  to be a reciprocal Lebesgue number function, for an open cover of , gives a rapid proof that is compact when it is complete, without recourse to sequential compactness or the Lebesgue covering lemma. Finally, we apply the same reasoning to another function to give sequential compactness.

Keywords: Pseudometric space, Cauchy sequence, total boundedness, Lebesgue number, Lebesgue covering lemma.