I came across the following while looking at some notes from a course about Sobolev spaces.
The exercise is to prove the density of compactly supported functions in the Sobolev space $W^{k,p}(\mathbb R^n).$ I am aware of a proof of this fact using mollifiers and cut-off functions. For some reason I am stuck to this. I suppose it is a straightforward application of the definition of convergence in Sobolev spaces combined with a pointwise estimate for the derivatives, arising when applying the Leibniz rule.
Let $\phi \in C^\infty (\mathbb R)$ such that
$$ \begin{cases} \phi(t)=0, & \text{if} \hspace{0.05in} t\leq -1 \\ % 0 \leq \phi(t) \leq 1, & \text{if} \hspace{0.05in} -1 \leq t \leq 0 \\ % \phi(t)=1 & \text{if} \hspace{0.05in} t \geq 0. \end{cases} $$
For $m>0$ define the function $ \displaystyle \phi_m(t)= \phi(m+t) \phi(m-t)$ and for $ x \in \mathbb R^n$ define $ \psi_m(x)= \phi_m(x_1) \cdots \phi_m(x_n).$
Given a function $ u \in W^{k,p}(\mathbb R^n) ,$ prove that $ \psi_m u \to u $ (as $ m \to \infty$ ) in the sobolev space $W^{m,p}(\mathbb R^n).$
Conclude that the functions with compact support are dense in $W^{m,p}(\mathbb R^n).$
Any help would be appreciated.
Thank you in advance.