They've proven that $E$ is continuous on $\mathbb R^{n+1}$. Now they consider $S\subset \mathbb R^{n+1}$ such that $E(c_0,\dots,c_n)\leq\Vert f\Vert_\infty +1$. I want to show that this set is bounded, but I'm not sure how to go about this. I tried proof by contradiction. So assume $S$ is not bounded. Then for each $M>0$, we have a $c\in\mathbb R^n$, such that $\Vert c\Vert>M$. I was thinking of maybe setting $M=\Vert f\Vert_\infty +1$, but that doesn't seem to guarantee that $E(c)>\Vert f\Vert_\infty+1$. So is a direct proof possible, and if so, how to go about this?