Proving that l^p is properly contained in l^q for 1<=p<q<=infty

[michael.wes]

Homework Statement

Prove that $$l^{p}\subsetneq l^q$$ for $$1\leq p < q \leq \infty$$.

Homework Equations

The Attempt at a Solution

I know how to prove the $$q=\infty$$ 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..

$$\sum_{n=1}^\infty |x_n|^q = \sum_{n=1}^\infty |x_n|^p|x_n|^{q/p}$$

I'm trying to show that $$\sum_{n=1}^\infty |x_n|^p = M^p$$ for some real, finite M implies that $$\sum_{n=1}^{\infty}|x_n|^q$$ 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

 

[micromass]

What about a simple comparison test for series?

 

[michael.wes]

Thanks micromass. that was a helpful tip

 

[Bacle2]

I was reviewing this, and this post came up. It may give someone else a slightly-different
perspective.Sorry if necro-posting is not helpful --please let me know:

A tip related to micromass': Think of the function f(x)= a^x for a constant;

where is it increasing/decreasing? Think , too, of a necessary condition for {a_n}

to be in l^p re the partial sums as n→∞.