I am trying to understand the proof in [1] for Theorem 2.3. In particular, there is a line with the following inequality
$\sum_{j=0}^p \frac{|t|^j}{j!} ||N||^j \le C_1(1 + |t|)^p$
with $C_1$ and $p$ somehow a function of $N$ has 1's on its superdiagonal, i.e.,
$$N = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 & 0 & 0\\\ 0 & 0 & 1 & \cdots & 0 & 0 & 0\\\\ \vdots & \vdots & \ddots & \cdots & \vdots & \vdots & \vdots \\\\ 0 & 0 & 0 & \cdots & 0 & 1 & 0 \\\\ 0 & 0 & 0 & \cdots & 0 & 0 & 1 \\\\ 0 & 0 & 0 & \cdots & 0 & 0 & 0 \end{pmatrix} $$
I have no idea where this inequality comes from. I found this discussion but it seems to have the same logical jump from a partial sum of an exponential to an inequality given some constant. Is there some general bound result that I am missing? Thanks!
[1] Sideris, Thomas C. Ordinary differential equations and dynamical systems. Vol. 2. Paris: Atlantis Press, 2013., link