If we are given that $O \subset \Omega$ is open and bounded and $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$. We have a sequence $\{u_{m}\}$ satisfying $$ u_{m} \rightharpoonup u\text{ } \text{ in }\text{ } W^{1,\infty}_{0}(O)$$ and
$$\limsup\limits_{m \rightarrow \infty}\int_{O} \langle a(x,u_{m}, \nabla u_{m}); \nabla(u_{m}-u)\rangle dx \leq 0$$ where $\langle \text{ } \rangle$ is the scalar product.
How does this imply the following two conditions to $u_{m} \rightharpoonup u$ in $W^{1,\infty}(O)$ and $\Vert \nabla u_{m} \Vert_{L^{\infty}(O)} \leq \sigma$ where $\sigma > 0$.
Please view this pdf link Research Paper Question Refers To. The above question refers to how we can use Lemma 1 at the start of the proof of Lemma 2.
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