Proof. Let $A$ be a stochastic matrix and let $m$ be an eigenvalue of $A$ , with $v$ eigenvector; then, for any self-consistent matrix norm $\left\Vert .\right\Vert $ , we have: $$ \left|m\right|\left\Vert v\right\Vert =\left\Vert mv\right\Vert =\left\Vert Av\right\Vert \leq\left\Vert A\right\Vert \left\Vert v\right\Vert , $$ that is, since $v$ is nonzero, $$ \left|m\right|\leq\left\Vert A\right\Vert . $$ Now, for a (doubly) stochastic matrix, $$ \left\Vert A\right\Vert _1 = \max_j \left(\sum_i \left|a_{ij}\right|\right)=1 $$ whence the conclusion.