Inequality on $\ell_p$: Proving or Disproving?

[forumfann]

Could anyone prove or disprove the following inequality:
$||x||_{p}\leq||x||_{p'}$ for all $x\in\mathbb{R}^{n}$ if $p'>p\geq1$?

By the way, this is not a homework problem.

Any help on this will be highly appreciated.

 

[berkeman]

Could anyone prove or disprove the following inequality:
$||x||_{p}\leq||x||_{p'}$ for all $x\in\mathbb{R}^{n}$ if $p'>p\geq1$?

By the way, this is not a homework problem.

Any help on this will be highly appreciated.

Where is the problem from?

 

[forumfann]

This a problem that I was curious about, because we know that $||x||_{m}\leq||x||_{1}$ for any positive integer $m$, and then I wondered if it is true for any $p\geq1$.

But it would be great if one can show the following:
$||x||_{p}\leq||x||_{1}$ for $p\geq1$,
so could anyone help me on this?

 

[trambolin]

$$\|x\|^p_p = \sum_i{|x_i|^p} \leq \left( \sum_i{(|x_i|)} \right)^p = \|x\|_1^p$$

Though I am very dizzy right now, it should be OK where I used $a^2+b^2 \leq (a+b)^2, a,b > 0$