This is on page 542 of Evans PDE book. The last inequality states that
$$\int_{U}{C(|Du|+1)|u|dx} \leq \frac{1}{2}\int_{U}|Du|^2dx + C\int_{U}{|u|^2+1 \ dx}$$
Where is this coming from? I think this is just young's inequality and then holder applied to $|Du|$ (since $u$ is assumed to be in $H_0^1[U]$) but why write it in such a weird way?
So we have the inequality $$ \int_U |Du |^2 + \mu |u|^2 dx \leqslant \frac{1}{2} \int_U |Du|^2 dx + C \int_U |u|^2 + 1 dx .$$ Now assume that $\mu$ is sufficiently large, for example let $\mu > C + \frac{1}{2} $ . Then this inequality becomes \begin{eqnarray*} \frac{1}{2}\int_U |Du|^2 dx &\leqslant& (C - \mu ) \int_U |u|^2 dx + C \int_U dx \\ &\leqslant & - \frac{1}{2} \int_U |u|^2 dx + C \int_U dx \end{eqnarray*} or $$ \frac{1}{2} \int_U |Du|^2 + |u|^2 dx \leqslant C \int_U dx .$$ Thus we can see that $$ \| u \|_{H_0^1 (U)} = \int_U |Du|^2 + |u|^2 dx \leqslant 2C \int_U dx \leqslant C' < \infty $$ holds.