91Ó°ÊÓ

Let \(S\) be a set and \(d\) a function from \(S \times S\) into \(\mathrm{R}^{1}\) with the properties: (i) \(d(x, y)=0\) if and only if \(x=y\). (ii) \(d(x, z) \leqslant d(x, y)+d(z, y)\) for all \(x, y, z \in S\). Show that \(d\) is a metric and hence that \((S, d)\) is a metric space.

Let \(A\) and \(B\) be compact sets in \(\mathbb{R}^{N}\) such that \(A \cap B=\varnothing\). Show that inf \(d(p, q)\). where the infimum is taken for all \(p \in A\), all \(q \in B\) is positive.