Let $(X,d)$ be a metric space with finite doubling constant $0<C_X$. It seems intuitive to me that if $\emptyset \subset Y\subseteq X$ then $Y$ is also doubling and it's doubling constant, 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?

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?