I was reading up and it says that $(\ell^2,\|.\|_2)$ is complete. I know that a metric space $X$ in which every Cauchy sequence converges to an element of $X$ is called complete. And I know that a sequence is Cauchy if given $\epsilon$, there exists $N \in \mathbb{N}$ such that $|x_m-x_n|<\epsilon$ for all $n,m>N$.
I was reading the proof that $(\ell^2,\|.\|_2)$ is complete, but I don't understand where does the $x_k^n$ comes from. What does it mean when it writes $x_k^n$?
Let $(x_n)$ be Cauchy in $\ell^2$, i.e. $\forall \epsilon>0$ there exists $N \in \mathbb{N}$ such that $\sum_{k=1}^\infty|x_k^n-x_k^m|^2 <\epsilon^2$ for $n,m>N$.
For any fixed $k_0$, $|x_{k_0}^n-x_{k_0}^m|<\epsilon$ for $n,m >N$. So $(x_{k_0}^n)$ is Cauchy in $\mathbb{K}$ ($\mathbb{R}$ or $\mathbb{C})$ and converges to say $y_{k_0}$.
Also, why did they square $\sum_{k=1}^\infty|x_k^n-x_k^m|^2 <\epsilon^2$?