Let $A \in \mathscr{M}_n(\mathbb{C})$. Show the equivalence of the following two assertions\\
(i) $A \in \mathbb{M}_n(v)$ for every sequence $v = (v_k)_{k \geqslant 0}$ of $\mathbb{C}$ satisfying $R_v > 0$.\\
(ii) $A$ is nilpotent (that is, there exists $k \in \mathbb{N}^*$ such that $A^k = 0_n$).