If $(X,d)$ is a metric space, then $diam(U_R(a))=2R$?

Question

Let $(X,d)$ be a metric space and $a\in X$. Does it hold that $diam(U_R(a))=2R$? Here $U_R(a):=\{x\in X : |x-a|<R\}$

I already showed this for the case that $(X,d)$ is a normed space. In that case it was easy to see that $diam(U_R(a))=Rdiam(U_1(a))=2R$.

However, I can't apply the same argument here. If dealing with real numbers this seems obvious, but I think this isn't true for general metric spaces...any ideas?

Answer 1

Think of a discrete space. Then the diameter of any subset with at least two points is $1$.