I would like to prove (B) below:
$\Omega$ is a bounded domain in $\mathbb{R}^n$, and $D$ denotes the gradient operator. Using the triangle inequality, one can readily obtain $$\int_{\Omega'}|f'(u_m(x))(Du_m)(x)-f'(u(x))(Du)(x)|\mathrm{d}x\leq\int_{\Omega'}|f'(u_m(x))|\cdot|(Du_m)(x)-(Du)(x)|\mathrm{d}x+\int_{\Omega'}|f'(u_m(x))-f'(u(x))|\cdot|(Du)(x)|\mathrm{d}x.$$ It will be great if we can show that $$|f'(u_m)(\partial_j u_m-\partial_j u)|\leq(\sup|f'|)|\partial_j u_m-\partial_j u|$$ for $j=1,2,\ldots,n$. But I don't know how to do that. Please help. Thank you.