How do you do the second part of question 8, chapter 5, of Evans' PDE book (first edition)? I have proven the inequality for smooth, compactly supported functions using integration by parts, and I understand why approximating sequences as described in the hint exist, but I can not use the hint to extend the inequality as required.
The question is
"Integrate by parts to prove the interpolation inequality
$\int_U |Du|^2\,dx\leqslant C\left(\int_Uu^2\, dx\right)^{\frac{1}{2}}\left(\int_U|D^2u|^2\,dx\right)^{\frac{1}{2}}$
for $u\in C^\infty_c(U)$. By approximation, prove this inequality if $u\in H^2(U)\cap H_0^1(U)$. "
The hint is to approximate $u$ by functions in $C^\infty_c(U)$ which converge to u in $H_0^1(U)$, and to approximate $u$ (also) by functions in $C^\infty(closure(U))$ which converge to $u$ in $H^2(U)$.