Trying to show that the matrix exponential $ \sum_{k=0}^{\infty} \frac{A^k}{k!} $converges because the sequence of partial sums $S_N=\sum_{k=0}^{N} \frac{A^k}{k!} $ is Cauchy.
It is Cauchy, if $$ \forall \varepsilon>0: \ \exists N_{\varepsilon}: \ N,\ M>N_{\varepsilon} \Longrightarrow \Vert S_M-S_N \Vert <\varepsilon. $$
So is this equivalent to showing that $$ \lim_{M,N \rightarrow \infty} \Vert S_M-S_N \Vert=0?$$