Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $(M,d)$ be a compact metric space and $f: M \rightarrow M$ a continuous function. I'm trying to prove that if $d(f(x),f(y)) \geq d(x,y)$ for every $x,y \in M$, then $f$ is an isometry. This is how far I could get: Since $f$ is a continuous function, the set $\{(x,y) \in M \times M : d(f(x),f(y)) = d(x,y)\}$ is closed, and hence compact. By symmetry, it is of the form $N \times N$. Thus, $N$ (with the metric induced by $M$) is a compact metric space such that $d(f(x),f(y)) = d(x,y)$ for every $x,y \in N$. I already know that this implies that the restriction of $f$ to $N$ is an isometry of $N$, but how to prove that the open set $M \setminus N$ is empty?

share|improve this question
Hint: suppose $x,y\in M$ are such that $d(fx,fy)>d(x,y)$. By induction, get a sequence of points $z_n$ such that $d(z_n,y)>d(z_{n-1},y)$ for all $n$. Combine with compactness of $M$. –  wildildildlife Jul 14 '11 at 13:06
A good exercise as a follow-up: prove that $f$ must also be surjective. –  Mark Jul 14 '11 at 16:18

1 Answer 1

up vote 2 down vote accepted

Look at the 12th post. Here

(This post is actually not correct and/or incomplete. The subsequences of a_n and b_n which converge due to compactness of the space do not necessarily have the same sub-indices, and so we cannot necessarily use the same function g(n) for both subsequences. The idea is in the right direction, but this does not suffice as it is)

share|improve this answer
that's a nice argument (the one you linked) –  Zarrax Jul 14 '11 at 14:50
You can find a common subsequence: Find $h(n)$ so that $a_{h(n)}$ converges. The sequence $b_{h(n)}$ has a convergent subsequence, which can be defined by the function $g(n)$, and $a_{g(n)}$ converges since it is a subsequence of a convergent sequence. –  yasmar Feb 21 '12 at 7:58
It seems everything is beyond my reach! Let me ask the Community why the added paragraph was not written as a comment first. That was a very naive way of down voting, also. –  Ehsan M. Kermani Feb 21 '12 at 9:32

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.