Question about convergence in $\mathcal D(\Bbb R)$

Question

Let $\phi\in \mathcal D(\Bbb R)$. How to prove or disprove convergence of $\phi_n(x)=\frac{1}{n} \phi(nx)$ in $\mathcal D(\Bbb R)$?

I tried to do this by definition (we have to check two conditions of convergence):

(1) Because of $\operatorname{supp} \phi_n\subset\frac{1}{n}\operatorname{supp}\phi$, we have that $supp \phi_n$ is limited and closed (by the definition of $\operatorname{supp}\phi_n$) and therefore, $\operatorname{supp}\phi_n=K$, $K$ is a compact in $\Bbb R$.

Now, I am not sure how to check the following second condition.

(2) How to prove or disprove uniform convergence of $\phi^{(\alpha)}_n$ on compact $K$?

Any help is welcome.

Answer 1

Hint: look at the sequence of derivatives.