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¿How can I prove, using a sequence of mollifiers, that the set $C^{\infty}_{c}(I)$ (also known as $\mathcal{D}(I)$) is dense in $W^{1,p}_{0}(I)$?

Also I want to know that if $u\in W^{1,p}(I)\cap C_{c}(I)$ then $u\in W^{1,p}_{0} (I)$

Here $I\subset\mathbb{R}$ a open interval.

¿Could help me by computing an unconventional example?

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    What is your definition of $W_0^{1,p}(I)$? I ask, because sometimes it is defined as the closure of $\mathcal{D}(I)$ w.r.t. the Sobolev norm. – 2017-01-25
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    $W^{1,p}_{0}(I)$ is the clousure of $C^{1}_{c}(I)$ in the space $W^{1,p}(I)$ w.r.t the Sobolev norm. $\vert\vert u\vert\vert_{L^{p}}+ \vert\vert u^{\prime}\vert\vert_{L^{p}} $ – 2017-01-25
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    In this case, I try to demonstrate, under this new definition, that the set of test functions is dense in $W^{1.p}_{0}(I)$. I try to demonstrate your definition. – 2017-01-25

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