Let $(X,d)$ be a metric space, $A\subset X$ and $0<r$. If $d(p,q)\leq 2r$ for all $p,q\in A$, is there some $x\in X$ such that $A$ is contained in the closed ball with center $x$ and radius $r$?
Motivation: Fix some $0<\epsilon$ and consider some Cauchy sequence. We know that $d(x_m,x_n)\leq \epsilon$ for all $m,n\geq N(\epsilon)$ for a Cauchy sequence and I was wondering if we can fit the tail of the sequence into a ball with diameter (not radius) equal to $\epsilon$.