This exercise is 6.19 in Rudin's Functional analysis.
The Problem:
$\Lambda \in \mathscr{D}'(\Omega), \ \phi \in \mathscr{D}(\Omega), \ (D^{\alpha}\phi)(x)=0, \ \forall \ x$ in the support of $\Lambda$ and multi-index $\alpha$, show that $\Lambda \phi=0$
Similar to Theorem 6.25 in Rudin (The proposition is that a distribution whose support is one point $p$ is a linear combination of $\delta$ and its derivatives at $p$), I can prove the case that the support of $\Lambda$ is compact, but I don't know how to generalize to the general case.
Thank you very much!