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I want to show that $W_0^{1,2}(a,b)$ is the closure of $C_c^{\infty}(a,b)$ by proving that for every $u \in W_0^{1,2}(a,b) ~\exists~ (u_n)_{n \in \mathbb{N}} \in C_c^{\infty}(a,b)$ so that $||u_n - u||_{W_0^{1,2}}$ converges to $0$.

But every explanation of $W_0^{1,2}$ just defines it as the closure of $C_c^{\infty}$. Does anyone know how to actually show this if the definition of $W_0^{1,2}(a,b)$ includes all functions of $W^{1,2}(a,b)$ with $f(a)=f(b)=0$?

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    This kind of magic is usually done by convolution with smooth mollifiers, see https://en.wikipedia.org/wiki/Mollifier In your case you need a mollifier $\phi$ with compact support. Then you need to modify the function $u$ you want to approximate (e.g by multiplying it with a smooth cuttoff function) in an $\varepsilon$-neighbourhood of the boundary to $u_\varepsilon$ so that the convolution with the scaled $\phi_\varepsilon$ is still in $(a,b)$. Then let $\varepsilon \rightarrow 0$ and estimate the $W^{1,2}$ norm of $u_\varepsilon\star \phi_\varepsilon - u$ – 2017-01-08
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    Damn, typo: The _support of the convolution_ with the scaled $\phi_\varepsilon$ should be still in $(a,b)$. – 2017-01-08

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