Can any metric on $\Bbb R^n$ be bounded above and below for any other metric? Let $d_1(x,y)$ and $d_2(x,y)$ be any two metrics on $\mathbb{R}^n$. Can it be shown that,
$$c\cdot d_2(x,y) \le d_1(x,y) \le C\cdot d_2(x,y)$$
for all $x,y \in \mathbb{R}^n$ for some fixed positive constants $c,C$? If not, under what conditions could such a relation hold (for a compact set it seems straightforward)? If yes, does this result hold for two arbitrary topological metric spaces as well?
Thanks!

If $d_1$ is the usual metric and $d_2$ is the discrete metric, then there are no such constants.
However, your statement is true if the metrics are induced by norms in $\mathbb{R}^n$.
