I am reading a paper and bumped at this lemma which I do not know the proof and would like to see some reference. Please suggested me a possible reference.
Let $M$ be a metric space and $K^{(n)}={x^n_1,...,x^n_k}$ denote a sequence of subsets of cardinality $\leq$ k in $M$. There is group acting on M by isometry such that the action is cocompact.(I do not know if the last fact is required for the lemma)
Lemma: After passing to a subsequence in the sequence $(K^{(n)})$ we can break $K^{(n)}$ as the disjoint unioun of nonempty subsets $K^{(n)}=\cup_{i=1}^l K^{(n)}_i$ so that 1. diam($K^{(n)}_i)\leq D$, where is a finite number and for all i=1,..,l,$n\in N$ and 2. $lim_{n\rightarrow \infty}$ dist ($K^{(n)}_i,K^{(n)}_j$)= $\infty$ for $i\neq j$