We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. The sequence $\left( M ^ { k } \right)$ converges to a stochastic matrix $B$ all of whose rows are equal to $P ^ { \infty } = \left( b _ { 1 } , \ldots , b _ { n } \right)$.
Prove that $P ^ { \infty }$ is the unique probability distribution $P$ invariant by $M$, that is, satisfying $P M = P$.