Let $A$ be a matrix of order $ n \times n$ and let $ \lim_{ k \rightarrow \infty} A^k = L > 0.$ I want to show that there exists some power $m$ such that $A^m>0$ and $ A^{m+i} > 0$ for any $ i \geq m.$ By a matrix $L>0$ and $A>0$ I mean all entries of the matrices are positive. Any hint or a proof please?
Thank you in advance.