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- #1

- Jan 26, 2012

- 995

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**Problem**: Let $(X,d)$ be a metric space. If $f:X\rightarrow X$ satisfies the condition

\[d(f(x),f(y))=d(x,y)\]

for all $x,y\in X$, then $f$ is called an

*isometry*of $X$. Show that if $f$ is an isometry and $X$ is compact, then $f$ is bijective and hence a homeomorphism.

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**Hint:**

Remember to read the POTW submission guidelines to find out how to submit your answers!