Let (X,d) be a metric space and A be a closed subset of X. Let B be a compact subset of X such that A$\cap$B=$\phi$. Prove that there exist a $\epsilon$$\gt$0 such that d(a,b)$\ge$$\epsilon$ for all a$\in$A and b$\in$B.
I think first I have to show that if b is not in A then there exist a $\epsilon$ such that d(a,b)$\ge$$\epsilon$. And then for each b$\in$B this holds and somehow use the compact ness of B. But I don't understand how to show this. Especially how to use the compact ness of B here. Thank you.
