I'm learning about Hilbert spaces and operators theory, from some book. I came across the following question -
And the books' answer:
What I don't understand in the proof -
Why can we understand that each sequence is Cauchy?
Moreover, why the following inequallity is true?
If $\sup_n|x_n^{(k)}-x_n^{(m)}|\leq\epsilon$ then it follows that for each $n\in\mathbb{N}$ we have $|x_n^{(k)}-x_n^{(m)}|\leq\epsilon$, because this expression is not bigger than the supremum on $n$. It follows that $(x_n^{(k)})_{k=1}^\infty$ is Cauchy for each $n$. (Cauchy with respect to the usual metric in $\mathbb{C}$).
As for the second question: the first inequality is the triangle inequality, the second follows from the fact that $x_j^{(k)}\to x_j$, the third follows from the definition of supremum norm.