This question comes from Introduction of theory of distribution by F.G.Friedlander (2Ed, Chap 1, Question 1.6).
Let $u \in \mathscr{D}'(\mathbb{R}^n)$ have the property that $\langle u,\phi\rangle\ge0$ for all real valued nonnegative $\phi\in C_c^\infty(\mathbb{R}^n)$. Show that $u$ is of order $0$ .
I tried to use contradiction. Its detail is below.
Fix $u \in \mathscr{D}'(\mathbb{R}^n)$ such that $\langle u,\phi\rangle\ge0$ for all real valued nonnegative $\phi\in C_c^\infty(\mathbb{R}^n)$.
There is a test function sequence $\{\phi_i\}\subset C_c^\infty(K),K\subset\subset\mathbb{R}^n$. Which has bounded $C(K)$ norm. In the meantime, $\lim_{i\to\infty}|\langle u,\phi_i\rangle| = +\infty$.
Because $u \in \mathscr{D}'(\mathbb{R}^n)$, so for this $K$, $\exists m\in\mathbb{N}$ and $C(K,m)\in\mathbb{R}$ s.t. $\forall i:|\langle u,\phi_i\rangle|\le C(K,m)\sum_{|\alpha|\le m}||\partial^\alpha\phi_i||_{C(K)}$.
So $\lim_{i\to\infty}C(K,m)\sum_{|\alpha|\le m}||\partial^\alpha\phi_i||_{C(K)} = \lim_{i\to\infty}\sum_{0<|\alpha|\le m}||\partial^\alpha\phi_i||_{C(K)}= +\infty$.
And here is where I got stuck. I can't combine "$\lim_{i\to\infty}\sum_{0<|\alpha|\le m}||\partial^\alpha\phi_i||_{C(K)}= +\infty$" with "$\langle u,\phi\rangle\ge0$ for all real valued nonnegative $\phi\in C_c^\infty(\mathbb{R}^n)$".
Any hint is super grateful. Thanks in advance!