real analysis – The convolution of a compactly supported distribution with a Schwarz function is a function of Schwarz

This is a question of Folland Chapter 9.

I would like to show that if $$psi in mathcal S$$, $$G in mathcal E$$(all of the compact support distribution), then $$G * psi in mathcal S$$.

Note that $$G$$ has a unique extension to a continuous linear function on smooth functions, and $$G * psi: = langle G, tau_x tilde psi rangle$$, or $$tau_x tilde psi (y) = psi (x-y)$$.

I'm trying to use the continuity of $$G$$ get

$$| G * psi (x) | = | langle G, tau_x tilde psi row | leq c sum_ {| alpha | leq N} || tau_x tilde psi || _ {[m,alpha]}$$
But I can not extract any information about $$| partial ^ alpha G * psi (x) |$$ show $$G * psi$$ is a function of Schwarz.

Any help would be appreciated.