Theorem. Let the matrix $A\in \mathbb{R}^{n\times n}$. Then $\left \| e^{A} \right \|\leq e^{\left \| A \right \|}$.
where $\left \| \cdot \right \|$ is a norm in $\mathbb{R}^{n\times n}$.
I'm trying to prove the theorem, starting from $S_{n}=\sum_{k=0}^{n}\frac{A^{k}}{k!}$ and $e^{A}=\sum_{k=0}^{\infty }\frac{A^{k}}{k!}$,doing the respective operations I arrive to
$\left \| e^{A} -S_{n}\right \|\leq \sum_{k=n+1}^{\infty }\frac{\left \| A \right \|^{k}}{k!}$.
I can not conclude the demonstration. Thank you very much for your help
Apply the last inequality for $n=−1$. Note that a sum with zero terms is zero.
Also, for this all to work your norm needs to be sub-multiplicative, like the operator-norms associated to vector space norms on $\Bbb R^n$.