I am trying to show that $u\ast\psi=v$ for some $v\in\mathcal{C}_c^{\infty}$ in the following way:
Step 1: Show that one can define $\langle u\ast\psi,\phi\rangle$ for $\phi\in\mathcal{S}'$
Step 2: Define a candidate for $v$ by using $\delta$-distributions for $\phi$.
Step 3: Show that $u\ast\psi=v$ by taking any $\phi\in\mathcal{S}$ and applying both sides to $\chi_{\epsilon}\ast\phi$ and letting $\epsilon\rightarrow0$.
This question is the exercise 2.2 (4) on the 11th page of Hintz's notes on microlocal analysis, which can be found at https://people.math.ethz.ch/~hintzp/notes/micro.pdf
Can anyone please suggest how to solve the question by following the above steps, which are hints of the exercise? Thanks a lot.