Properties of doubling metric spaces

Question

At present I work with tools that involves doubling metric space, my definition of DME is:

A metric space $X$ is called doubling with constant $N$, where $N \geq 1$ is an integer, if, for each ball $B(x , r )$, every $\frac{r}{2}$-separated subset of $B(x , r )$ has at most $N$ points. A $\frac{r}{2}$-separated subset is a set of points $\{x_i\}$ such that $d(x_i,x_j) \geq \frac{r}{2}$ for $i \neq j$.

I have been studying and trying to prove some results but with the following two I haven't had much success:

1. if $X$ is a metric space such that every open ball of radius $r > 0$ in $X$ can be covered by $M$ open balls of radius $\frac{r}{2}$ then $X$ is doubling with constant $M^2$.

For these I considerer a $\frac{r}{2}$ separated set in a ball $B(x,r)$ For this result try to give a set $\frac{r}{2}$ separated in the ball $B(x,r)$ and try to see its intersections with the cover that by hypothesis exists, also I had the doubt if in the statement I should have "at most M balls".

2. Let $X$ be a doubling metric space with constant $N$ and let $k \geq 1$ be an integer. Then every $2^{−k} r$ -separated set in every ball $B(x , r )$ in $X$ has at most $N^k$ points.

For this result I have tried to use the following characterization of the doubling metric spaces: If $X$ is a doubling metric space with constant $N$ then every open ball of radius $r > 0$ in $X$ can be covered by $N$ open balls of radius $r/2$

but I have not been able to conclude anything in both results any hint would be much appreciated. Also if you could recommend me books to consult the proofs of the results I would appreciate it.

Answer 1

maybe Lemma 2.3 in A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa by T. Hytönen can be of help. It shows the equivalence of different definitions of "geometrically-doubling" metric space, in particular the third one is about the maximum number of $\delta$-separated points in a ball of radius 1.