Let $(X,d)$ be a metric space with finite Assouad dimension $0<C\_X$. It seems intuitive to me that if $\emptyset \subset Y\subseteq X$ then $Y$ is also doubling and its Assouad dimension, denoted here by $C\_Y$, should satisfy $C\_Y\leq c C\_X$ (where $c$ is some absolute constant independent of $X$ and of $Y$).


Is this true, and if so where can I find this fact?



Monotonicity of doubling dimension
This is Lemma 9.6(i) in J. C. Robinson, *Dimensions, embeddings, and attractors.* Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011.


In the proof the author says "it is obvious". I am no longer sure if it is obvious (perhaps it is) since I see a potential issue: if $Y\subset X$, then every ball in $Y$ is a restriction of a ball from $X$, but if $B$ is a ball in $X$ not centered at $Y$, then $B\cap Y$ is not a ball in $Y$ so there are less balls in $Y$ to cover. Is it an issue? I haven't checked (I still did not have my morning coffee).


