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Could you help me with this problem?

Let $d$ be a metric on $[0,1]$ consistent with the standard topology. Prove that the metric space: $([0,1], d)$ has at most one isometry (except for identity).

I would really appreciate your help.

Thank you.

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Notice that it is enough to show that an isometry which fixes the endpoints is the identity.

Suppose then $f:[0,1]\to[0,1]$ is an isometry such that $f(0)=0$, and let $x\in[0,1]$. Notice that $f$ is strictly increaing —because it is continuous for the usual topology and injective.

It is easy to check that the sequence $(f^n(x))_{n\geq1}$ is monotone, so that it converges with respect to the usual topology of $[0,1]$. The hypothesis then implies that it also converges with respect to $d$ and, in particular, it satisfies the condition of Cauchy. Since $f$ is an isometry, $d(f^n(x),f^{n+1}(x))$ does not depend on $n$ because $f$ is an isometry, and then the Cauchy condition implies that this distance is zero: in particular, $d(x,f(x))=0$, that is $x=f(x)$.

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  • $\begingroup$ It may be a stupid question, but how do we know that the sequence $(f^n(x))_{n\geq1}$ is monotone? $\endgroup$
    – Andrew
    Apr 22 '13 at 17:19
  • $\begingroup$ Think about it for a while. I'll add the details in a few hours if you did not see it on your own. $\endgroup$ Apr 22 '13 at 17:25
  • $\begingroup$ I still don't see that. I don't think it somehow results from the fact that isometry preserves distance, does it? I've tried various combinations with distance between $f^{n}, \ f^{n-1}, \ f(x), \ x$, but it doesn't seem to be the right approach. $\endgroup$
    – Andrew
    Apr 22 '13 at 18:03
  • $\begingroup$ Could you post the details now, please? $\endgroup$
    – Andrew
    Apr 23 '13 at 5:03
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Hint: Continuity is a topological property.

Spoiler: The following is the shape of a proof.

  1. Isometries are continuous functions.
  2. Continuity is a topological property.
  3. Isometries are bijections.
  4. Continuous bijections $([0,1],d_\textrm{Eucl.}) \to ([0,1],d_\textrm{Eucl.})$ are strictly increasing or strictly decreasing.
  5. A strictly increasing bijective function $[0,1]\to[0,1]$ which is an isometry w.r.t. the metric $d$ is unique.
  6. Similarly for a strictly decreasing bijective function.

Edit: For point 5, suppose $f,g$ both have this property. Then consider $h=f^{-1}\circ g$. This is a continuous bijection (check), so is strictly increasing/decreasing. Suppose $h(x)\neq x$. Then $d(h^n(x),h^{n-1}(x)) = d(h(x),x)\neq 0$. But let $n\to\infty$; by monotonicity, $h^n(x)\to x_0$, and by continuity, $d(h^n(x),h^{n-1}(x)) \to 0$. Contradiction.

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  • $\begingroup$ Could you tell me a bit more about point 5 ? $\endgroup$
    – Andrew
    Apr 22 '13 at 17:20
  • $\begingroup$ Whoops, just realized this is more or less equivalent to the other answer. $\endgroup$ Apr 22 '13 at 17:47

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