I have the following problem:
Let $\phi(x) \in C_0^\infty(\mathbb{R}^n)$ satisfy $\phi \geq 0$ and $\phi(0) = 1$. Show that $\phi_n = \phi/n$ converge to $0$ in $\mathcal{D}'(\mathbb{R}^n)$.
My solution sketch is the following. $\lim_{n->\infty} \int_{\mathbb{R}^n} \frac{\phi(x)}{n} f(x)dx = \lim_{n->\infty} \frac{1}{n}\int_{\mathbb{R}^n} \phi(x) f(x)dx \leq \lim_{n->\infty} \frac{C}{n} = 0$ where I in the last step have used that $\phi$ has compact support so that the integral will be finite.
I'm a little confused about this solution since I have not used the addional properties of $\phi$ and therefore I think I have cheated a little. So my question is then what I have missed in my solution?
Thanks in advice!