Let $\phi:\mathbb{R}^n \rightarrow \mathbb{R}$ with $\phi(x):=\exp(1/(|x|^2-1))$ for $|x|<1$ and $\phi(x):=0$ otherwise and $\phi_{\varepsilon}(x)=1/\varepsilon^n\cdot\phi(x/\varepsilon)$. Show that $\phi_\varepsilon$ is in ${C_c}^\infty(\mathbb{R}^n)$.
I know how to show it if $n=1$, but how can I show it for a multidimensional function? (Maybe induction?) I would be glad if you gave me some tips.