- #1

michael.wes

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## Homework Statement

Prove that [tex]l^{p}\subsetneq l^q[/tex] for [tex]1\leq p < q \leq \infty[/tex].

## Homework Equations

## The Attempt at a Solution

I know how to prove the [tex]q=\infty[/tex] case, and how to find an example where the containment is proper (just take the harmonic series to the appropriate power), but I cannot find a way to prove the actual containment. I tried re-writing as follows, to try and use the cauchy-schwarz inequality, but I don't even know if it holds in an infinite dimensional vector space..

[tex]\sum_{n=1}^\infty |x_n|^q = \sum_{n=1}^\infty |x_n|^p|x_n|^{q/p}[/tex]

I'm trying to show that [tex]\sum_{n=1}^\infty |x_n|^p = M^p[/tex] for some real, finite M implies that [tex]\sum_{n=1}^{\infty}|x_n|^q[/tex] is bounded above by something which depends on this limit, and hence I can just raise both sides to the power 1/q, and I will have the containment. But like I said before, I have searched around but I could not find any hints.

Any help is appreciated!

M