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Let $A$ be a square matrix. To show: Matrix exponential converges to some matrix $X$.

$$ \lim_{N \rightarrow \infty} \sum_{k=0}^{N}\frac{A^k}{k!} =X $$

In some proofs that I have seen it is stated that because (for a sub-multiplicative norm) $$ 0 \le \sum_{k=0}^{\infty} \left\Vert \frac{A^k }{k!} \right\Vert \le  \sum_{k=0}^{N} \frac{\Vert A \Vert ^k }{k!} then the series $\sum_{k=0}^{N}\frac{A^k}{k!}$ has to be convergent. That however isn't clear to me.

To me more intuitive way to show convergence would be to show that $$ \lim_{N \rightarrow \infty} \left\Vert \sum_{k=0}^{N} \frac{A^k}{k!} -X \right\Vert  =0$$ and use some intuitive matrix norm for which it is clear that all elements of $\frac{A^k}{k!} -X$ converge to zero.

Any hints?