We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. The sequence $\left( M ^ { k } \right)$ converges to a stochastic matrix $B = \left( \begin{array} { l l l } b _ { 1 } & \cdots & b _ { n } \\ b _ { 1 } & \cdots & b _ { n } \\ b _ { 1 } & \cdots & b _ { n } \end{array} \right)$ all of whose rows are equal. We denote by $P ^ { \infty }$ the row $\left( b _ { 1 } , \ldots , b _ { n } \right)$. Prove that, $\forall i \in \llbracket 1 , n \rrbracket$, $b _ { i } > 0$.
We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. We set $\varepsilon = \min _ { 1 \leqslant i , j \leqslant n } m _ { i , j }$. The sequence $\left( M ^ { k } \right)$ converges to a stochastic matrix $B = \left( \begin{array} { l l l } b _ { 1 } & \cdots & b _ { n } \\ b _ { 1 } & \cdots & b _ { n } \\ b _ { 1 } & \cdots & b _ { n } \end{array} \right)$ all of whose rows are equal. We denote by $P ^ { \infty }$ the row $\left( b _ { 1 } , \ldots , b _ { n } \right)$.
Prove that, $\forall i \in \llbracket 1 , n \rrbracket$, $b _ { i } > 0$.