In Euclidean space, any $R$-ball can be covered by $C$ $\frac{R}{2}$-balls where $C$ is independent of $R$.
But in hyperbolic space, this does not hold.
Here we have a definition :
Def : A metric space is doubling if there is constant $C$ s.t. any ball $B$ can covered by at most $C$ balls whose radius is half the radius of $B$.
Problem : A metric space is doubling iff any $R$-ball is covered by at most $C$ subsets whose diameter is equal to $R$ or smaller than $R$.
Proof : We have a claim that if $E$ has a diameter $R$, then it can be covered by $C'$ $\frac{R}{2}$-balls. How can we prove this ?