I'm trying to understand the proof of the trace theorem. This is from a course I am taking, so I will write out what we have done explicitly.
$\textbf{Trace Theorem}$ Suppose $\Omega \subset\mathbb{R}^n$ is bounded, open with $C^1$ boundary. Then there exists a bounded linear operator $Tr:W^{1,p}(\Omega )\rightarrow L^p(\partial\Omega )$. Further, if $u\in C(\overline\Omega )\cap W^{1,p}(\Omega )$ then $Tru=\overline{u}$ for all $x\in\partial\Omega$, where $\overline{u}$ denotes the uniformly continuous extension of $u$ to $\overline{\Omega}$.
To prove this, we need the extension theorem:
$\textbf{Extension Theorem}$ Suppose $\Omega \subset\mathbb{R}^n$ is bounded, open with $C^1$ boundary. Suppose further that $\overline{\Omega}\subset V$ where $V\subset\mathbb{R}^n$ is bounded. Then there exists a bounded linear operator $E:W^{1,p}(\Omega )\rightarrow W^{1,p}_0(V)$ such that $Eu = u$ a.e. for all $x\in \Omega$. Further, if $u\in C(\overline\Omega )\cap W^{1,p}(\Omega )$ then $Eu \in C(\overline{V})$.
Now we are in a position to prove the theorem. I won't go in to too much detail but the idea is standard. We prove the first part of the theorem for test functions. We then have the result for $Eu$ by density, so I define $Tr(u) = \lim _{n\rightarrow \infty}Tr(u_n)$ where $(u_n)$ is my approximating sequence of test functions for $Eu$ and $Tr(u_n):={u_n|}_{\partial \Omega}$
Now suppose $u$ is uniformly continuous. We want to show the two constructions coincide. Firstly, the extension theorem tells me that $Eu$ is a uniformly continuous extension of $u$ to $\overline{\Omega}$, and since this extension is unique we have that $Eu=\overline{u}$ on the boundary. So I need to show that $Eu=Tr(u)$ on the boundary. We know that $J_\epsilon\ast u\rightarrow u$ uniformly on $\partial\Omega$ as $\epsilon\rightarrow 0$, where $J_\epsilon$ is the standard mollifier. This is where I am stuck. In the notes it says that $J_\epsilon\ast u$ also converges in $L^p(\partial\Omega )$ and therefore this limit must be $Tr(u)$. Why? Why isn't it just $Eu$ again?
Any help much appreciated!