Hausdorff metric inherits completeness

Theorem 1.

If (X,d) is a complete metric space, then the Hausdorff metric induced by d is also completePlanetmathPlanetmathPlanetmathPlanetmath.


Suppose (An) is a Cauchy sequencePlanetmathPlanetmath with respect to the Hausdorff metric. By selecting a subsequence if necessary, we may assume that An and An+1 are within 2-n of each other, that is, that AnK(An+1,2-n) and An+1K(An,2-n). Now for any natural numberMathworldPlanetmath N, there is a sequence (xn)nN in X such that xnAn and d(xn,xn+1)<2-n. Any such sequence is Cauchy with respect to d and thus convergesPlanetmathPlanetmath to some xX. By applying the triangle inequalityMathworldMathworldPlanetmath, we see that for any nN, d(xn,x)<2-n+1.

Define A to be the set of all x such that x is the limit of a sequence (xn)n0 with xnAn and d(xn,xn+1)<2-n. Then A is nonempty. Furthermore, for any n, if xA, then there is some xnAn such that d(xn,x)<2-n+1, and so AK(An,2-n+1). Consequently, the set A¯ is nonempty, closed and boundedPlanetmathPlanetmathPlanetmath.

Suppose ϵ>0. Thus ϵ>2-N>0 for some N. Let nN+1. Then by applying the claim in the first paragraph, we have that for any xnAn, there is some xX with d(xn,x)<2-n+1. Hence AnK(A¯,2-n+1). Hence the sequence (An) converges to A in the Hausdorff metric. ∎

This proof is based on a sketch given in an exercise in [1]. An exercise for the reader: is the set A constructed above closed?


  • 1 J. Munkres, TopologyMathworldPlanetmathPlanetmath (2nd edition), Prentice Hall, 1999.
Title Hausdorff metric inherits completeness
Canonical name HausdorffMetricInheritsCompleteness
Date of creation 2013-03-22 14:08:51
Last modified on 2013-03-22 14:08:51
Owner mps (409)
Last modified by mps (409)
Numerical id 8
Author mps (409)
Entry type TheoremMathworldPlanetmath
Classification msc 54E35
Related topic Complete