We assume that $A$ is strictly positive and diagonalizable over $\mathbb{C}$. For all $Y \in \mathcal{M}_{n,1}(\mathbb{R})$, for all $p \in \mathbb{N}^*$, we denote $Y_p = \left(\frac{A}{\rho(A)}\right)^p Y$. Let $\lambda \in S = \operatorname{sp}(A) \setminus \{\rho(A)\}$. Let $Y \in \ker\left(A - \lambda I_n\right)$. Show that the sequence $\left(Y_p\right)_{p \in \mathbb{N}^*}$ converges to 0.