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Can you provide an example of a sequence in $l_4$ but not in $l_3$?

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    $\begingroup$ Hint: Try to think about the harmonic series. $\endgroup$
    – user99914
    Nov 20 '15 at 13:01
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    $\begingroup$ What about $\frac{1}{n^{1/3}}$? $\endgroup$
    – Crostul
    Nov 20 '15 at 13:03
  • $\begingroup$ That is an extremely good example. $\endgroup$ Nov 20 '15 at 14:02
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Hint. We have $x := (\frac 1n) \in \bigcap_{p > 1}\ell^p \setminus \ell^1$. Now consider $(x_n^q)$ for the right $q$.

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