Metric Space Diameters of Sets: Find Condition

[Kindayr]

Homework Statement

Find a condition on a metric space $(X,d)$ that ensures that there exist subsets $A,B$ of $X$ with $A\subset B$ such that $diam(A)=diam(B)$.

Homework Equations

$diam(A)=\sup\{d(r,s):r,s\in A\}$;
$A\subseteq B\implies diam(A)\leq diam(B)$.

The Attempt at a Solution

Well I know examples of where this is true (ie, let $A=(-\infty,5]\cup [-5,\infty)\subset (-\infty,4]\cup [-4,\infty)=B$). But I don't know which condition allows this to be true. Any help is good. Thank you!

 

[Kindayr]

Bump. Anyone? Any Idea?