Prove that l^p is a subset of l^q for all p,q from 1 to infinity

[cbarker1]

Homework Statement

Prove that l^p is a subset of l^q for all p,q from 1 to infinity. Then prove it is strict subset. First, prove that a^t<=a for all t,a in (0,1]. Then prove that finite sum of |x_i|^t<= the sum of |xi|.

Relevant Equations

a^t<=a for all a,t
p-norm's definition.

Dear everyone,

I am having trouble with this problem. I have convinced myself that the $a^t-a\leq 0$ is true. Now, I am trying to applying this inequality for the finite series and I don't know where to start. After that, proving that the p-norm is less or equal to the q-norm.

Thanks,
Cbarker1

 

[pasmith]

Is there a condition on $p$ and $q$, such as $q < p$? Otherwise you are being asked to prove $l^p \subsetneq l^q \subsetneq l^p$ which is impossible.

If you have two sequences of non-negative numbers, with the property that each element of the first sequence is less than or equal to the corresponding element of the second sequence, what can you say about the sums of those sequences?