I am trying to show that $W^{1,2,0}(U)$ is dense in $H^1(U)$ for an open, bounded set $U\subset\mathbb{R}.$ Where $W^{1,2,0}(U)$ is defined to be the space of smooth functions $f:U\to\mathbb{C}$ such that $f'$ exists and are in $L^2(U)$ and the norm is given by $\|f\|_{H^1(U)}=\|f\|_2+\|f'\|_2$
I have something written, but it's incomplete. If someone could look it over and give me any help, I would be grateful.
I have proved before that $(g_\epsilon*f)'=g_\epsilon*f'$ (*)
Where $g\in C_c^\infty{(U)}$ is a nonnegative valued function with the properties that $\int_U g(x)dx=1$ and for each $\epsilon>0$ define $g_\epsilon(x)=\frac{1}{\epsilon}g(\frac{x}{\epsilon})$ then for each $f\in L^2(U)$ we define $(g_\epsilon*f)=\int_{\mathbb{R}}g_\epsilon(x-y)f(y)dy$.
I also have that $g_\epsilon*f$ is well-defined, smooth, and that $\lim\limits_{\epsilon\to0}\|g_\epsilon*f-f\|_2=0. (**)$
So what I want to do is prove that if $f\in H^1(U)$ then there exists a sequence of functions $(f_m)_{m=1}^\infty\in H_0^1(U)$ such that $\|f_m-f\|_{H^1(U)}\to 0$ as $m\to\infty.$
Using the above, since $f\in H^1(U)$ we have that $f$ and $f'\in L^2(U).$
Thus if I define $f_\epsilon:=g_\epsilon*f$ and $h_\epsilon:=g_\epsilon*f'$ then by $(**)$ above I have that $f_\epsilon\to f$ and $h_\epsilon\to f'$ in $L^2(U)$ as $\epsilon\to 0.$
Furthemore by $(**)$ i have that $f'_\epsilon=(g_\epsilon*f)'=g_\epsilon*f'=h_\epsilon.$
So $\|f_\epsilon-f\|_{H^1(U)}=\|f_\epsilon-f\|_2+\|f'_\epsilon-f'\|_2=\|f_\epsilon-f\|_2+\|h_\epsilon-f'\|_2\to 0$ as $\epsilon\to 0$. If we take $\epsilon=\frac{1}{m}$ then we have the desired result IF $f_\epsilon$ are in $W^{1,2,0}(U)$ but I'm not sure how to show this last part.