Suppose I have a set $\displaystyle X$ equipped with two equivalent metrics, $\displaystyle d_1$ and $\displaystyle d_2$, meaning that the metrics induce the same topology on $\displaystyle X$. I know that $\displaystyle d_1$ and $\displaystyle d_2$ need not be comparable on all of $\displaystyle X$, however, is it true that they will be comparable on compact subsets of $\displaystyle X$?