Uniform convergence of $A^n/n!$

Question

In a proof regarding finite space Markov Jump Processes in which the function $P(t)=e^{tG}$ is a solution to both the backward and forward Chapman-Kolmogrov equations, one of the steps assumes that for all $m \times m$ matrices $A$, the series \begin{equation} \sum_{n=0}^{\infty} \frac{A^n}{n!} \end{equation} converges uniformly. Does anybody have a link to a proof of this as it seems to be absent from my notes. Thanks guys!

Answer 1

The series converges uniformly on subsets of $M_{n}(\mathbb{C})$ such that $\left\|A\right\|\le R$ for some $R>0$. Then uniform convergence is guaranteed by the uniform convergence of the power series for $e^{x}$ for sets on which $|x|<R$.