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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.

snowman
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1 Answers1

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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.

zhw.
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