Let $f$ be a smooth bounded function, and suppose moreover that $f \in W^{k,2} (\mathbb{R^n})$ for all $k \geq 0$.
Fix a natural number $k_0$. I want to show that there exists a sequence of Schwartz functions $f_n$ such that $f_n \to f$ in $W^{k_0,2}$ and pointwise everywhere.
I tried:
By molification, I can show that there exists such a sequence consisting of merely smooth $f_n$.
I tried to truncate $f_n$ to get Schwartz functions, but then I cannot control the $W^{k_0 ,2 }$ norms.
How should I proceed?
Choose $k_1 > \max(k_0, \frac{n}{2})$. There exists a sequence $f_n \in \mathcal{S}$ such that $f_n \to f$ in $W^{k_1,2}$. Therefore (by standard embedding) this convergence is also in $W^{k_0,2} \cap L^\infty$.