From a real-valued function f, unbounded on a totally bounded subset of a metric space, we construct a Cauchy sequence in S on which f is unbounded. Taking f to be a reciprocal Lebesgue number function, for an open cover of S, gives a rapid proof that S 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 f to give sequential compactnessKeywords: Pseudometric space, Cauchy sequence, total boundedness, Lebesgue number, Lebesgue covering lemmaJournal of the Nigerian Association of Mathematical Physics, Volume 20 (March, 2012), pp 1 – 4