Let A and B be nonempty compact subsets of $\mathbb{R}^n$ with $A \cap B = \emptyset$. Then there exists a $\delta > 0 $ such that for all $a \in A$ and $b \in B$, $|a-b|> \delta$.
I am having trouble with this problem because I do not see where the compactness comes in. To me it seems that since $|a-b|$ is a metric we have that it must be greater than or equal to zero. Therefore if we assume the negative, i.e $|a-b|=0$ then by defintion of a metric we must have $a=b$ which is a contradiction to the fact that A and B are disjoint. What is wrong and how should I go about this. Thank you!