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

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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?

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(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.$

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  • $\begingroup$ Thank you, it is so clear, what about the 3 lines after equation(3), I also did not understand them, could u clarify them please, if you have time? $\endgroup$ – Intuition Oct 16 '17 at 18:10
  • $\begingroup$ Did the author used the idea of "increasing sequence and bounded above so it is convergent" ? $\endgroup$ – Intuition Oct 16 '17 at 18:16

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