Limit of distribution functions

Question

Let $D$ be a space of distribution functions. I need to prove that for any function $F \in D'$ there are distribution functions $F_n \in D$ for which $F_n(f) \rightarrow F(f)$ for every $f \in D$.
I know how to prove $D$ is dense in $D'$ in weak-$*$ topology, but i don't know what to do next. I've tried to construct a sequence of $F_n$ using density of $D$, but my teacher said that we can't do this because $D'$ topology is not metrizable. Would be glad for any help.

Answer 1

Take $\psi \in \mathcal{D}$ such that $\int \psi = 1$ and let $\psi_n(x) = n \psi(nx).$ Then $\psi_n \to \delta$ is a mollifier.

Also take $\rho \in \mathcal{D}$ such that $\rho(0)=1$ and let $\rho_n(x) = \rho(x/n).$ Then $\rho_n \to 1$ pointwise and in $\mathcal{E} = C^\infty.$

Let $F_n = \rho_n (\psi_n*F).$

Does $F_n \in \mathcal{D}$? Does $F_n \to F$ in $\mathcal{D}'$?