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Let $\{v_n\}_{n \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over $\mathbb{C}$ such that $\{v_n\}_{n \in \mathbb{N}}$ is linearly independent and $v_n \to v_0$.

I would like to know if exist a subsequence $\{v_{n_k}\}_{k \in \mathbb{N}}$ such that for each fixed $p \in \mathbb{N}$: $$ v_{n_p} \notin \overline{span} \{v_{n_k}\}_{k > p} $$

Thanks.

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    $\begingroup$ The answer is still obviously "no", e.g. fix $v_0 \neq 0$ and let $v_n = (1 - 1/n)v_0$. $\endgroup$
    – Nik Weaver
    Commented Mar 13, 2018 at 20:27
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    $\begingroup$ Better on MathStackExchange $\endgroup$
    – paul garrett
    Commented Mar 13, 2018 at 20:34
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    $\begingroup$ The question now seems reasonable. Voting to reopen. $\endgroup$
    – Nik Weaver
    Commented Apr 7, 2018 at 14:38
  • $\begingroup$ ok thanks @NikWeaver it was just a missing assumption, an involuntary error $\endgroup$
    – Matey Math
    Commented Apr 7, 2018 at 18:05
  • $\begingroup$ the assumption that $v_n\to 0$ does not contibute to anything, it seems. The span does not depend on the norm of the generating vecotrs. $\endgroup$
    – erz
    Commented Apr 7, 2018 at 22:10

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Clearly the answer is "no" if the sequence is constant. Did you mean to put some other assumptions?

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  • $\begingroup$ thanks for your comment, now i edit and add other assumption $\endgroup$
    – Matey Math
    Commented Mar 13, 2018 at 20:18

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