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Let $1 \leq p < q < \infty$. Show that if $x=(a_k)∈ℓ^p$

i.e. the condition that the series $$\sum_{k=1}^\infty|a_k|^p$$ converges holds, then $x∈ℓ^q$

i.e. $$\sum_{k=1}^\infty|a_k|^q$$ converges.

I did try to use the ratio test but I don't think it will work because of the power on the terms in the series.


marked as duplicate by user296602, Did real-analysis Jan 27 '16 at 18:18

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  • $\begingroup$ What do $p$ and $q$ have to do with each other? $\endgroup$ – user296602 Jan 27 '16 at 17:46
  • $\begingroup$ sorry fixed now $\endgroup$ – snowman Jan 27 '16 at 17:47
  • $\begingroup$ Why the deliberate self-duplicate? math.stackexchange.com/q/1628081 $\endgroup$ – Did Jan 27 '16 at 18:19

Because $\sum_{n=1}^{\infty} |a_n|^p < \infty,$ $|a_n|^p \to 0.$ Thus there exists $N$ such that $|a_n|^p< 1$ for $n\ge N.$ For such $n$ we can say that, since $q/p > 1,$

$$|a_n|^q = (|a_n|^p)^{q/p}\le |a_n|^p.$$

Thus $\sum_{n=1}^{\infty} |a_n|^q$ converges by the comparison test.


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