Suppose \(A, B \subseteq \RR^n\) are disjoint and compact. Prove that there exist \(a\in A, b\in B\) such that \[ \norm{a - b} = \inf\theset{\norm{x-y} \suchthat x\in A,\, y\in B} .\]
Solution
Define a function \[ d: A \cross B &\to \RR \\ (x, y) &\mapsto \norm{x- y} .\] Then \(d\) is a continuous function on a compact topological space (where the product is compact by Tychonoff), and the extreme value theorem applies: \(d\) attains its min/max for some pair \((a, b)\) in its domain.
Note that disjointness just guarantees that \(\norm{a-b}>0\), since \(\norm{a-b} = 0 \implies a=b\) and \(A \intersect B = \emptyset\).