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Shorter Notes: A Characterization of Metric Completeness

J. D. Weston
Proceedings of the American Mathematical Society
Vol. 64, No. 1 (May, 1977), pp. 186-188
DOI: 10.2307/2041008
Stable URL: http://www.jstor.org/stable/2041008
Page Count: 3
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Shorter Notes: A Characterization of Metric Completeness
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Abstract

A proof is given of a theorem, relevant to fixed-point theory, which implies that a metric space $(X, d)$ is complete if and only if, for each continuous function $h: X \rightarrow \mathbf{R}$ bounded below on $X$, there is a point $x_0$ such that $h(x_0) - h(x) < d(x_0, x)$ for every other point $x$.

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