Hausdorff metric inherits completeness

Theorem 1.

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

Proof.

Suppose $(A_{n})$ is a Cauchy sequence with respect to the Hausdorff metric. By selecting a subsequence if necessary, we may assume that $A_{n}$ and $A_{n+1}$ are within $2^{-n}$ of each other, that is, that $A_{n}\subset K(A_{n+1},2^{-n})$ and $A_{n+1}\subset K(A_{n},2^{-n})$ . Now for any natural number $N$ , there is a sequence $(x_{n})_{n\geq N}$ in $X$ such that $x_{n}\in A_{n}$ and $d(x_{n},x_{n+1})<2^{-n}$ . Any such sequence is Cauchy with respect to $d$ and thus converges to some $x\in X$ . By applying the triangle inequality, we see that for any $n\geq N$ , $d(x_{n},x)<2^{-n+1}$ .

Define $A$ to be the set of all $x$ such that $x$ is the limit of a sequence $(x_{n})_{n\geq 0}$ with $x_{n}\in A_{n}$ and $d(x_{n},x_{n+1})<2^{-n}$ . Then $A$ is nonempty. Furthermore, for any $n$ , if $x\in A$ , then there is some $x_{n}\in A_{n}$ such that $d(x_{n},x)<2^{-n+1}$ , and so $A\subset K(A_{n},2^{-n+1})$ . Consequently, the set $\overline{A}$ is nonempty, closed and bounded.

Suppose $\epsilon>0$ . Thus $\epsilon>2^{-N}>0$ for some $N$ . Let $n\geq N+1$ . Then by applying the claim in the first paragraph, we have that for any $x_{n}\in A_{n}$ , there is some $x\in X$ with $d(x_{n},x)<2^{-n+1}$ . Hence $A_{n}\subset K(\overline{A},2^{-n+1})$ . Hence the sequence $(A_{n})$ 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?

References

1 J. Munkres, Topology (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	Theorem
Classification	msc 54E35
Related topic	Complete