If $1\leq p < r \leq \infty$ , then $\| x \|_p \geq \| x \|_r $

Question

For $1\leq p \leq \infty$ , let $\| \cdot \|_p $ be the $\mathcal{\ell_p}$ norm on $\mathbb{R}^n$. I need to show that if $1\leq p < r \leq \infty$ , then $\| x \|_p \geq \| x \|_r $.

I have tried Holder's inequality and Minkowski inequality here but they didn't solve my purpose. I am completely stuck at this point.

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Answer 1

Something from a $\LaTeX$ file I have somewhere, posting it here as "Community Wiki" answer for further use, in case it proves helpful.

Theorem. For any sequence $x=(x_1,\dots,x_n)\in\mathbb{R}^n$, $p > 0 \mapsto \lVert x\Vert_p$ is non-increasing. In particular, for $0 < p \leq q <\infty$, $$ \left(\sum_i \lvert{x_i}\rvert^q\right)^{1/q} = \lVert x\Vert_q \leq \lVert x\Vert_p = \left(\sum_i \lvert{x_i}\rvert^p\right)^{1/p}\;. $$

Proof (sketch). To see why, one can easily prove that if $\lVert x\Vert_p = 1$, then $\lVert x\Vert_q^q \leq 1$ (bounding each term $\lvert{x_i}\rvert^q \leq \lvert{x_i}\rvert^p$), and therefore $\lVert x\Vert_q \leq 1 = \lVert x\Vert_p$. Next, for the general case, apply this to $y \stackrel{\rm def}{=} \frac{x}{\lVert x\Vert_p}$, which has unit $\ell_p$ norm, and conclude by homogeneity of the norm.

(The above deals with $x\neq 0$. Note that the case $x=0$ is immediate.)

Finally, the case $q=\infty$ is straightforward, as for any $p < \infty$ and any $x\in\mathbb{R}^n$ we have $\lVert x\rVert_\infty = \max_i \lvert{x_i}\rvert = (\max_i \lvert{x_i}\rvert^p)^{1/p} \leq \left(\sum_{i} \lvert{x_i}\rvert^p\right)^{1/p} = \lVert x\rVert_p.$