Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r < \rho$. Show that $\| \cdot \| _ { r }$ is a norm on $\mathscr { D } _ { \rho } ( \mathbb { R } )$ and that $\| f g \| _ { r } \leqslant \| f \| _ { r } \cdot \| g \| _ { r }$ for all $f , g \in \mathscr { D } _ { \rho } ( \mathbb { R } )$.