Calculus on manifolds

Okay, so I'm reading Spivak's "Calculus on manifolds" and in exercise 1-21b you have to prove that in an euclidean space if A is a closed set, B is compact and their intersection is empty, there is d>0 such that |y-x|>=d for every x in A and y in B. It also says this doesn't work if B is closed but not compact. Now, no matter how I approach this it seems to me that the sets only need to be closed. Anybody knows what I'm doing wrong?

ITT high school kid thinks he is a genius and can unravel the mysteries of the universe by asking questions, on 4chan, to which if he got the correct answer he would not understand it until his 2nd year of graduate school. 
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