I'm going through the proof of Corollary 8.11 in Brezis' Functional Analysis, Sobolev Spaces and Partial Differential Equations which states:
Let $G \in C^1(\mathbb{R})$ be such that $G(0) = 0$, and let $u \in W^{1,p}(I)$ with $p \in [1, \infty]$, then $G \circ u \in W^{1,p}(I)$ and $(G \circ u )' = (G' \circ u)u'$.
Following his proof and filling some steps I've shown that $(G \circ u)$ and $(G' \circ u)u'$ are in $L^p(I)$, focusing on the case $p < \infty$ we obtain $(u_n)_n \subseteq C_c^\infty(\mathbb{R})$ such that $u_{n\vert I} \to u$ in $W^{1,p}(I)$ (and hence in $L^\infty(I)$) then he states
Thus $(G\circ u_n)_{\vert I} \to G \circ u$ in $L^\infty(I)$ and $(G' \circ u_n)u'_{n\vert I} \to (G' \circ u)u'$ in $L^p(I)$.
I've managed to prove the first assertion but not the second. So far i've gotten that (to make the notation lighter I'm omitting the restriction to $I$ but it should be there)
\begin{align*} ||(G' \circ u_n)u'_n - (G' \circ u)u'||_{p} &\leq ||(G'\circ u_n)u'_n -(G' \circ u)u'_n||_p + ||(G' \circ u)u'_n - (G' \circ u)u'||_p\\ &\leq ||u'_n||_p ||G' \circ u_n - G' \circ u||_\infty + ||G'\circ u||_{\infty}||u'_n-u'||_{p} \end{align*} since $u'_{n \vert I} \to u'$ in $L^p(I)$ we have that $\sup_{n \in \mathbb{N}}||u'_{n \vert I}||_{p}$ is finite and since $G' \circ u$ is bounded a.e. the second summand vanishes, so it'd be enough to prove $G' \circ u_n \to G' \circ u$ in $L^\infty(I)$, I was thinking on using that $u_n \to u$ uniformly a.e. alongside the continuity of $G'$ but since $I$ could be unbounded that doesn't necessarily work.
