How do I show convergence in a normed space? I know a sequence is convergent if $$\forall \epsilon \gt 0 \, \exists n_\epsilon \in \mathbb N \, \forall n \geq n_\epsilon :\|x-x_n\| \lt \epsilon$$
But I am unsure how to apply this... Specifically in the following example:
Let $A_{n, m} := \begin{cases} \frac{1}{n+m+1} & \text{if $m \leq n$} \\ 0 & \text{otherwise.} \end{cases} $
Let $k \in \mathbb{N}$; consider the sequence $(a_n)_{_{n \in \mathbb{N}}}$ in $\mathbb R^k$ with $$a_n := (A_{n,0}, ..., A_{n,k-1}) \in \mathbb R^k$$
Is $(a_n)_{_{n \in \mathbb{N}}}$ in $(\mathbb R^k, \|{\cdot}\|_\infty)$ convergent?
I also don't quite understand what $\|{\cdot}\|_1$, $\|{\cdot}\|_2$ and $\|{\cdot}\|_\infty$ etc mean.
I feel like I am generally quite shaky in this entire topic. Any help with this example would help me a lot to understand what I am supposed to do.