Suppose that the sequence of vectors $\left( P ^ { ( k ) } \right) _ { k \in \mathbb { N } }$ converges to a vector $P = \left( p _ { 1 } , \ldots , p _ { n } \right)$. Show that $P T = P$, that for all $i \in \llbracket 1 , n \rrbracket$, $p _ { i } \geqslant 0$ and that $p _ { 1 } + \cdots + p _ { n } = 1$.