Let $A\in L(\mathbb R^n)$, and set $C_j=\sum\limits_{i=0}^j \frac {A^i}{i!}$.
I'm trying to show that $C_j$ is a Cauchy sequence wrt the operator norm.
So far I have $||C_m-C_n||=||\sum\limits_{i=n+1}^m\frac{A^i}{i!}||\leq\sum\limits_{i=n+1}^m\frac{||A||^i}{i!}$
In finite dimensional spaces evey linear operator is bounded.
So $||A|| \leq M$ for some $M>0$
The series $\sum_{i=n+1}^m\frac{M^i}{i!} \to^{m,n \to +\infty} 0$ by convergence of the series,for every $M>0$
So the sequence $C_j$ is Cauchy.