Equivalence of $p$-Norms on Finite-Dimensional Linear Spaces

Question

Let $1\leq p,q\le\infty$, let $\mathbb F=\mathbb R$ or $\mathbb C$, and let $d$ be a positive integer. Is it possible to show the equivalence of the norms $\left\|\cdot\right\|_p$ and $\left\|\cdot\right\|_q$ on $\mathbb F^d$ without resorting to Hölder's inequality?

Answer 1

Corollary 13.29. of this file shows this directly.