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Let $1 \leq p \leq \infty$ and $1 \leq q \leq \infty$ be two intgers

Is it true that $p<q \Longrightarrow \ell^p \subset \ell^q$

Thanks

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  • $\begingroup$ I think you mean $\ell^q\subset\ell^p$ $\endgroup$
    – Jason Born
    Jun 12, 2017 at 17:15
  • $\begingroup$ yes, now i edited $\endgroup$
    – Matey Math
    Jun 12, 2017 at 17:16
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    $\begingroup$ The answer is: yes, it is true. $\endgroup$
    – Jason Born
    Jun 12, 2017 at 17:16
  • $\begingroup$ thanks @JasonBorn $\endgroup$
    – Matey Math
    Jun 12, 2017 at 17:18
  • $\begingroup$ duplicate of (math.stackexchange.com/q/4094). $\endgroup$
    – Jean Marie
    Jun 12, 2017 at 19:11

1 Answer 1

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Yes. To see this note that if $\sum |a_n|^p < \infty$ then it must be that $|a_n|^p< 1$ for sufficiently large $n$. Hence since $\frac{q}{p} >1$ it follows that $|a_n|^q<|a_n|^p$, which implies the result

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    $\begingroup$ thanks @Christopher Halverson for your answer $\endgroup$
    – Matey Math
    Jun 12, 2017 at 17:20

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