Let $H_0^1(0,1) = \overline{\mathcal{D}(0,1)}$, with the usual Sobolev-norm.
Show:
$h \in C^1[0,1], h(0)=h(1)=0$. Then $h \in H^1_0(0,1)$.
I know that $h \in H^1(0,1)$. Also i know that there exist some sequences $f_k,g_k \in \mathcal{D}(0,1)$ such that $\|h-f_k\|_{L^2} \rightarrow 0, \|h'-g_k\|_{L^2} \rightarrow 0$.
To show $h \in H^1_0(0,1)$, i have to somehow establish that it is possible to choose sequences $f_k,g_k$ which satisfy $(f_k)' = g_k$.
Thats where i'm stuck. I also know that for the last step i have to use $h(0)=h(1)=0$ somehow, because for example the function $h = 1$ is in $C^1[0,1]$ but not in $H^1_0(0,1)$...
Let $g_k \in \mathcal{D}(0,1)$ converge to $h'$ in $L^{2}$. Since $\int_0^{1} h'(t)dt =h(1)-h(0)=0$ we may choose $g_k$'s with $\int_0^{1} g_k(t)dt=0$ for each $k$.
[I will let you figure out why this is possible. Hint: Consider $g_k-(\int_0^{1}g_k(t)dt)\phi$ for a suitable $\phi \in \mathcal D(0,1)$].
Define $f_k(x)=\int_0^{x} g_k(t)dt$. Then, $f_k'=g_k$. Also, $f_k \in \mathcal D(0,1)$. Now, $h(x)=\int_0^{x}h'(t)dt$ because $h(0)=0$. Note that $|\int_0^{x}h'(t)dt-\int_0^{x}g_k(t)dt|\leq \int_0^{1}|g_k(t)-h'(t)|dt\leq \sqrt {\|g_k-h'\|_2}$. Hence, $f_k \to h$ uniformly, which certainly implies $f_k \to h$ in $L^{2}$.