I've proven that for $1 \leq p < q \leq \infty$, then $\|x\|_q \leq \|x\|_p \leq n^\frac{1}{p}\|x\|_q$. How can I use this to show that if a set $A\subseteq\mathbb{R}^n$ is open with respect to the $d_2$ metric if and only if it is open with respect to the $d_p$ metric for $1 \leq p \leq \infty$?
One idea I have is to compare two balls in $A$: $B^2_r(a) = \{x\in A|d(x,a) < r_2\}$ (for the $d_2$ metric), and $B^p_r(a) = \{y\in A|d(y,a) < r_p\}$ (for the $d_p$ metric). I'm not sure where to go from here, though; should I try to show that for $p > 2, B^2_r(a) \subseteq B^p_r(a)$ and vice-versa for $p \leq 2$? Or perhaps I should compare the radii $r_2$ and $r_p$? I'm starting to think it's not so helpful to compare two balls centered around the same point...