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.