Let $u \in L^p(\Omega)$ with $1<p \leq \infty$. Proposition 9.3 states two characterization of a function being in $W^{1,p}(\Omega)$. One is that, if $f\in W^{1,p}(\Omega)$, then there exists $C>0$ for every $K \subset \Omega$ that is compact, for all $h\in \mathbb R^n$ such as $|h|<\text{dist}(K,\partial\Omega)$ we have $$\Vert\tau_hf-f\Vert_{L^\infty(K)}\le \| \nabla f\|_{L^\infty(\Omega)} |h|$$ where $\tau_hf(x)=f(x+h)$.
The proof is done first for $C_{c}^{\infty}$ functions, then by approximation of $W^{1,p}(\Omega)$ by them. Let me rewrite Brezis' proof below, and then let me ask my question.
Proof: Assume first that $u \in C_c^{\infty}\left(\mathbb{R}^N\right)$. Let $h \in \mathbb{R}^N$ and set $$ v(t)=u(x+t h), \quad t \in \mathbb{R} . $$
Then $v^{\prime}(t)=h \cdot \nabla u(x+t h)$ and thus $$ u(x+h)-u(x)=v(1)-v(0)=\int_0^1 v^{\prime}(t) d t=\int_0^1 h \cdot \nabla u(x+t h) d t . $$ It then follows that for $1 \leq p<\infty$, $$ \left|\tau_h u(x)-u(x)\right|^p \leq|h|^p \int_0^1|\nabla u(x+t h)|^p d t $$ and $$ \begin{aligned} \int_\omega\left|\tau_h u(x)-u(x)\right|^p d x & \leq|h|^p \int_\omega d x \int_0^1|\nabla u(x+t h)|^p d t \\ & =|h|^p \int_0^1 d t \int_\omega|\nabla u(x+t h)|^p d x \\ & =|h|^p \int_0^1 d t \int_{\omega+t h}|\nabla u(y)|^p d y . \end{aligned} $$ If $|h|<\operatorname{dist}(\omega \partial \Omega)$, there exists an open set $\omega^{\prime} \subset \subset \Omega$ such that $\omega+t h \subset \omega^{\prime}$ for all $t \in[0,1]$ and thus $$ \left\|\tau_h u-u\right\|_{L^p(\omega)}^p \leq|h|^p \int_{\omega^{\prime}}|\nabla u|^p . $$
This concludes the proof of (ii) for $u \in C_c^{\infty}\left(\mathbb{R}^N\right)$ and $1 \leq p<\infty$. Assume now that $u \in W^{1, p}(\Omega)$ with $1 \leq p<\infty$. By Theorem 9.2 there exists a sequence $\left(u_n\right)$ in $C_c^{\infty}\left(\mathbb{R}^N\right)$ such that $u_n \rightarrow u$ in $L^p(\Omega)$ and $\nabla u_n \rightarrow \nabla u$ in $L^p(\omega)^N \forall \omega \subset \subset \Omega$. Applying (4) to $\left(u_n\right)$ and passing to the limit, we obtain (iii) for every $u \in W^{1, p}(\Omega)$, $1 \leq p<\infty$. When $p=\infty$, apply the above (for $p<\infty$ ) and let $p \rightarrow \infty$.
My question is if it would have been possible to do it directly for $W^{1,p}(\Omega)$ noticing that their "restriction on lines" (that is, the function $v$) is absolutely continuous. Thanks!