Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $\mathcal{M}_n(\mathbb{Z})$ the set of $n \times n$ matrices with integer coefficients.
Suppose that $M \in B_n \backslash \mathcal{M}_n(\mathbb{Z})$. Deduce that there exists a nonzero matrix $Q$ and $\epsilon > 0$ such that $\{M + tQ, t \in [-\epsilon, \epsilon]\} \subset B_n$, and conclude that every vertex of $B_n$ is of the form $P^\sigma$.