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As written, the question is: Let (X,d) be a compact metric space. Prove that for each $\epsilon>0$ there exists a positive integer $N$ such that for each $S \subseteq X$, if $S\thicksim Z_N$, then there exists $p,q \in S$ such that $d(p,q)< \epsilon$.

My question is-- isn't this kind of trivially true?, For any epsilon, chose N=1. Then if there is a bijection from S to {1}, there exists p and p in S such that $d(p,p)=0< \epsilon$. (The question never said p and q had to be distinct). Except in this case, X doesn't even need to be compact. I feel like I am missing something in the way the question is phrased. Either it is intended that p and q must be distinct, or I am misinterpreting what the question is asking.

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What does $S\sim Z_N$ mean? – Cameron Buie Nov 2 '12 at 9:10
up vote 1 down vote accepted

It is poorly stated: the intention is that the points $p$ and $q$ be distinct. Here’s a correct restatement:

Let $\langle X,d\rangle$ be a compact metric space. Then for each $\epsilon>0$ there is an integer $n>1$ such that whenever $S\subseteq X$ and $|S|=n$, there are $p,q\in S$ such that $0<d(p,q)<\epsilon$.

Here’s a hint for the proof: start by covering $X$ with open balls of radius $\frac\epsilon2$.

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