Show the Euclidean metric and maximum metric are strongly equivalent.
Work in $\mathbb R^2$ since the idea carries over easily to higher dimensions. Let $(x,y) \in \mathbb R^2$ and assume (with no loss of generality) that $|x| \le |y|$.
Since $|x|^2 + |y|^2 \le 2|y|^2$ you have $$\sqrt{x^2 + y^2} \le \sqrt 2 |y| = \sqrt 2 \max \{|x|,|y|\}.$$
Since $|x| \le |y|$ you have $$\max\{|x|,|y|\} = |y| = \sqrt{y^2} \le \sqrt{x^2 + y^2}.$$