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.