$p$-norms inequalities I got this math homework and I can't do it.
I have to prove the inequalities for p-norms in $\mathbb R^n$:


*

*$\|x\|_1\ge\|x\|_2\ge\|x\|_\infty$

*$\|x\|_1\le\sqrt{n}\|x\|_2$ (I have already proven this one.)

*$\|x\|_2\le\sqrt{n}\|x\|_\infty$

*$\|x\|_1\le\sqrt{n}\|x\|_\infty$


Can you help me with the others, please? Thank you very much!
 A: I will do the first one to get you started, and sketch the others.
Write $x = (x_1, x_2, \ldots x_n)$. Then:


*

*$\|x\|_{\infty} = \max_{1 \leq j \leq n} |x_j|$

*$\|x\|_2 = \sqrt{|x_1|^2 + |x_2|^2 + \cdots + |x_n|^2}$

*$\|x\|_1 = |x_1| + |x_2| + \cdots + |x_n|$


First, note that for any $1 \leq j \leq n$, we have $|x_j|^2 \leq |x_1|^2 + |x_2|^2 + \cdots + |x_n|^2$, and therefore, taking square roots, $|x_j| \leq \|x\|_2$. Since this is true for every $j$, it is true for whichever $j$ maximizes $|x_j|$, and therefore $\|x\|_{\infty} = \max |x_j| \leq \|x\|_2$.
Next, note that $|x_1|^2 + |x_2|^2 + \cdots + |x_n|^2 \leq (|x_1| + |x_2| + \cdots + |x_n|)^2$, since all of the terms on the left hand side appear on the right hand side (along with other nonnegative terms) when you expand the right hand side. Therefore, taking square roots, we get $\|x\|_2 \leq \|x\|_1$.
You already got (2), so I'll skip that one.
For (3), note that each $|x_j|$ is no larger than $\max_{1\leq j \leq n}|x_j| = \|x\|_{\infty}$, so $|x_1|^2 + |x_2|^2 + \cdots + |x_n|^2 \leq n (\|x\|_{\infty})^2$. It should be clear what to do next.
(4) is false as stated. However, the inequality $\|x\|_1 \leq n\|x\|_{\infty}$ is true, because as in (3), each $|x_j|$ is no larger than $\max_{1\leq j \leq n}|x_j| = \|x\|_{\infty}$, therefore $|x_1| + |x_2| + \cdots |x_n|$ is no larger than...?
