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
1. The function \(d: A \times B \to \RR\), \(d(x, y) = \|x - y\|\), is continuous.
the norm and subtraction are continuous.
2. \(A \times B\) is compact.
the product of two compact sets is compact (Tychonoff).
3. Hence \(d\) attains its minimum on \(A \times B\): there exist \(a \in A\), \(b \in B\) with \(d(a, b) = \inf_{x \in A, y \in B} d(x, y)\).
a continuous real-valued function on a compact set attains its infimum.
4. Q.E.D.
3.