Suppose that $ \sum_{m=0}^\infty A_m=A$, is this proof ok for $$\|A\|\leq \sum_{m=0}^\infty \|A_m\|$$ where $A\in\mathcal{M}_n$ is a matrix $n\times n$?
Proof: Define $S_k=\sum_{m=0}^k A_k$ then $$\|A\|\leq \|A-S_k\|+\|S_k\| \leq \|A-S_k\|+\sum_{m=0}^k\|A_m\| $$ then as $k\to \infty$ we get $$ \|A\|\leq \sum_{m=0}^\infty\| A_m\|$$ seems ok, but I'm using triangle inequality of $\|\cdot\|$ for finite terms, which I supposed holds.