Show that $\Sigma_{N}$ is a closed, bounded and convex subset of $\mathbb{R}^{N}$.
Where $\Sigma_{N}$ denotes the set of vectors $p \in \mathbb{R}^{N}$ such that $\sum_{i=1}^{N} p_{i} = 1$ and $p_{i} \geqslant 0$ for all $1 \leqslant i \leqslant N$.