Equivalence of p-norms It's known that in a finite-dimensional space $\mathbb{R}^n$ (where $n \geq 1$ is a natural number), for every $p,q \geq 1$ and $x \in \mathbb{R}^n$, where $p \leq q$ ,
$$
\left\| x\right\|_q \leq \left\| x\right\|_p \leq n^{\frac{1}{p} - \frac{1}{q}}\left\| x\right\|_q. 
$$
Where is this stated? What is a good reference/citation for this property of $p$-norms?
Please advise.
 A: Well, a possible reference is Wikipedia. There are probably lots of books on analysis where this relation is stated, but since it is very elementary, it might often be left as an exercise.
The first inequality follows from the fact that $y^q \leq y^p$ for all $y\in[0,1]$ and hence
\begin{align}
\| x \|_q &= \left(\sum_{i=1}^n | x_i |^q \right)^{1/q}  \\
&= \| x \|_p \left(\sum_{i=1}^n \left( \frac{| x_i |}{\| x \|_p}\right)^q \right)^{1/q} \\
&\leq \| x \|_p \left(\sum_{i=1}^n \left( \frac{| x_i |}{\| x \|_p}\right)^p \right)^{1/q} \\
&= \| x \|_p \left( \frac{\| x \|^p_p}{\| x \|^p_p} \right)^{1/q} \\
&= \| x \|_p .
\end{align}
The second inequality is an immediate consequence of Hölder's inequality, as 
\begin{align}
\| x \|_p &= \left( \sum_{i=1}^n | x_i |^p \cdot 1 \right)^{1/p} \\
&\leq \left( \! \left( \sum_{i=1}^n \left(| x_i |^p \right)^{q/p} \right)^{p/q} \cdot \left( \sum_{i=1}^n 1^{1/(1-p/q)} \right)^{1-p/q} \right)^{1/p} \\
&= n^{\frac{1}{p}-\frac{1}{q}} \, \| x \|_q.
\end{align}
