A Short Proof of Completion Theorem for Metric Spaces

Abstract

The completion theorem for metric spaces is always proven using the space of Cauchy sequences. In this paper, we give a short and alternative proof of this theorem via Zorn’s lemma. First, we give a way of adding one point to an incomplete space to get a chosen non-convergent Cauchy sequence convergent. Later, we show that every metric space has a completion by constructing a partial ordered set of metric spaces.