My text wants me to prove that $e^A$ converges for any $n \times n$ matrix and put a bound on it in terms of $|A|$ and $n$ where $| |$ is the norm induced by identification of $M(n, n)$ with $\mathbb{R}^{n^2}$. I can prove convergence and it's obviously bounded by $e^{|A|}$ but I don't see how to get it in terms of $n$.
To prove convergence, Just note that $e^{|A|} = \sum_{i=0}^{\infty}\frac{1}{i!}|A|^{i}$ converges. This means that
$ \sum_{i=0}^{j}\frac{1}{i!}|A^{i}| \le \sum_{i=0}^{j}\frac{1}{i!}|A|^{i} \le e^{|A|} $ This means that the partial sums of the norms are non-decreasing and bounded. They hence converge and the series is absolutely convergent which implies that it's convergent.
I can't get a bound on $|e^{A}|$ other than $e^{|A|}$ though. The problem says to bound it in terms of $|A|$ and $n$. I interpret this as proving that some function $f(|A|, n)$ exists with $|e^{A}| \le f(|A|, n)$ but I don't see how to do this. Any hints?