# A difficulty in understanding the proof of completeness of $l_{2}$.

A part of the proof is given in the following pictures:

But I do not understand why equation(2) & (3) is right, could anyone explain this for me please?

ِAlso, I did not understand the idea used in the last paragraph starting from $sup ||x_{n}|| = M,$ could anyone explain this for me?

(2): we have $\xi_k^{(n)} \to \xi_k$ as $n \to \infty$, hence
$\sum_{k=1}^j |\xi_k^{(n)}|^2 \to \sum_{k=1}^j |\xi_k|^2$ as $n \to \infty$.
(3) $\sum_{k=1}^j |\xi_k^{(n)}|^2 \le \sum_{k=1}^{\infty} |\xi_k^{(n)}|^2=||x_n||^2.$