I am trying to establish the compact embedding of $W^{1,1}_0(\Omega)$ into $L^1(\Omega)$ where $\Omega$ is a bounded open subset of $\mathbb{R}$.
I can mimmick the proof of the Rellich-Kondrachov theorem by mollifying a bounded sequence $\{u_i\}\subset W^{1,1}_0(\Omega)$, bounding their derivatives and aplying Arzelà-Ascoli, completing the argument by noting (on an un-labelled subsequence) $$ \|u_i-u_j\|_{L^1} \leq \|u_{i,\varepsilon}-u_i\|_{L^1} + \|u_{j,\varepsilon}-u_j\|_{L^1} + \|u_{i,\varepsilon} - u_{j,\varepsilon}\|_{L^1}, $$ and that $\|\cdot\|_{L^1}\lesssim_{\Omega}\|\cdot\|_{C^0}$ when $\Omega$ is bounded.
I was wondering if there was a more direct approach (without the detour of mollifying) by noting that $W^{1,1}_0(\Omega)$ functions are (modulo equality a.e.) absolutely continuous that vanish at the endpoints, so that $$ u(x) = \int_{a}^{x}u'(t)\,\mathrm{d}t. $$
I can easily bound $\|u\|_{C^0}\lesssim\|u\|_{W^{1,1}}$ from the above, but don't see how to bound $\|u'\|_{C^0}$ to conclude equicontinuity. I also tried to make $$ |u(x)-u(y)| \leq \int_{x}^{y}|u'(t)|\,\mathrm{d}t $$ small uniformly in $u'$, but this didn't seem promising either without a way to bound $u'$ outside of an integral.
Is this approach "viable"? If so, I would appreciate any hints (or suggestions of modification) to progress the strategy outlined above.