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 $x\_k^n$? 

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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 $\sum\_{k=1}^\infty|x\_k^n-x\_k^m|^2 <\epsilon^2$?