I came across these inequalities while learning about Schwartz functions (Classical Fourier Analysis, Grafakos) and I have no idea how to prove this:


For $x \in \mathbb{R}^{n}$ and $\alpha = (\alpha\_{1}, \ldots, \alpha\_{n}) \in \mathbb{N}^{n}$, we set


$$ x^{\alpha} = x\_{1}^{\alpha\_{1}}\cdots x\_{n}^{\alpha\_{n}}.$$


Then prove that there exists a constant $c\_{n,\alpha}$ such that



> 
> **$$\left| x^{\alpha}\right| \leq c\_{n,\alpha}|x|^{|\alpha|}$$**
> 
> 
> 


where $|\alpha| = \alpha\_{1} + \cdots + \alpha\_{n}$.


Conversely, for every $k \in \mathbb{N}$, there exists a $C\_{n,k}$ such that



> 
> $$|x|^{k} \leq C\_{n,k}\sum\limits\_{|\beta| = k}|x^{\beta}|$$
> 
> 
> 


Any help would be appreciated.


P.S. Please let me know if the question is too elementary for this forum.



An inequality involving multi-index
The first inequality with constant 1 follows from
$
\vert x^\alpha\vert\le\Vert x\Vert\_{\infty}^{\vert \alpha \vert},\quad\text{where $\Vert x\Vert\_{\infty}$ is the sup-norm.}
$


The second equality, also with constant 1, is due to $
\Vert x\Vert\_{\infty}^k=\max\_{1\le j\le n} \vert x\_j\vert^k\le \max\_{\vert \alpha\vert=k}
\vert x^\alpha\vert.
$


