In the book Functional Analysis of Rudin, the following theorem is proved: given a distribution $u$ and a sequence of test functions $\phi_n$ which converges to $\phi$, we have that $u *\phi_n $ converges to $u * \phi$ in the sense of $C^\infty$. We call $f_n$ converges to $f$ in the sense of $C^\infty$, if every partial derivative of $f_n$ uniformly converges to the correspondent partial derivative of $f$ on every compact set.
My question is: if the test function $\phi$ is fixed, and we are given a convergent sequence of distributions $u_n$ with limit $u$, do we have again the convergence of $u_n * \phi$ in $C^\infty$?
I think this is a very natural question. I have proved the pointwise convergence of $u_n * \phi$, and it seems $u_n * \phi$ converges in the sense of distribution too. I believe the convergence in $C^\infty$ is false, but I can't construct a counterexample. I would be very grateful if anyone could help.
Suggestion. Show that for a dense set of test functions $\psi$ (chosen to be independent of $n$),
$< u_n * \phi, \psi>= <u_n, \phi*\psi>$