Show that the $l^p$ norms with $1\leq p,q\leq \infty$ are equivalent

Question

I want to show that for any $1\leq p,q\leq \infty$, that there exists $a,b$ such that $a\left\|x\right\|_p \leq \left\|x\right\|_q \leq b\left\|x\right|_p$ for all $x$ in $\mathbb{R}^n$.
Is there a way to do this without Hölder's Inequalities?

Answer 1

$\|x\|_{\infty} \leq 1$ implies $\|x\|_{p} \leq n^{1/p}$ for any $p \in [1,\infty)$. Use this and a scaling argument to show that $\|x\|_{p} \leq n^{1/p} \|x\|_{\infty}$. [You have to consider $y=\frac x {\|x\|_{\infty}}$ when $x \neq 0$]. On the other hand, $ \|x\|_{\infty} \leq \|x\|_{p} $ because $|x_i| \leq \|x\|_{p} $ for each $i$. Hence all of the $p$-norms with $p<\infty$ are equivalent to $\infty$ norm. This implies that any two of the $p$-norms are equivalent. [ $\|x\|' \leq c\|x\|''$ and $\|x\|'' \leq d\|x\|'''$ implies $\|x\|' \leq cd\|x\|'''$ etc].