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Math Help - Prove that the little l(p) sequence spaces are complete.

  1. #1
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    Unhappy Prove that the little l(p) sequence spaces are complete.



    The book gave me the "hint" to mimic the proof of the l(1) space, but I can't find it anywhere. I have no idea how to approach this. I understand I want to show that every cauchy space converges to something in l(p), or that every sequence of sequences becomes arbitrarily close to another sequence. I know it follows from completeness on R, but thats all I know. I'm totally lost.
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  2. #2
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    ||x_n - x_m|| >= ||x_n_k - x_m_k||

    So, if (x_n) is a Cauchy sequence, then so are (x_n_k) for all k.
    Since R is complete every (x_n_k) converges against a number y_k.
    Now show that (x_n) converges against y:=(y_1,y_2,...) in the p-norm.
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  3. #3
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    Disagree.

    \|x\| = \big(\sum_{n=1}^{\infty}|x_n|^p\big)^{1/p}

    So if x_n = \{x_{n_k}\}_{k=1}^{\infty} and m similarly, then \|x_n - x_m\| may be greater than, less than, or equal to \|x_{n_k} - x_{m_k}\|

    Also, I'm pretty sure my way is the wrong way to approach this. I don't think we're interested in subsequences of x_n at all
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