Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote
$$J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$$
the supremum of $J_{f}$ on $\Sigma_{N}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$ the set of $p$ in $\Sigma_{N}$ for which the supremum is attained.
Show that $\Sigma_{N}(f)$ is non-empty.