Let $A \in M_p(\mathbb{C})$ be nilpotent of order $k$. Let $B \in M_p(\mathbb{C})$. We assume that $A$ and $B$ commute and that $E(B)$ exists.
We admit that, for every integer $i$ between 1 and $p$,
$$\lim_{n \rightarrow \infty} \left(I_p + \frac{1}{n}B\right)^n = \lim_{n \rightarrow \infty} \left(I_p + \frac{1}{n}B\right)^{n-i}$$
Show that $E(A + B)$ exists and that $E(A + B) = E(A)E(B)$.