Exercise 15, § 5

Let S, T two finite sets in $\mathbf{R}^n$ with no common points, and such that $\mathrm{Card}(S \cup T) \geq n + 2$. In order that there exists a hyperplane separating S strictly from T, it is necessary and sufficient that for every finite set $F \subset S \cup T$ of $n + 2$ points, there exists a hyperplane separating $F \cap S$ strictly from $F \cap T$ (use Helly’s th. (II, p. 68, exerc. 21)). Show that in this statement we cannot replace the number $n + 2$ by $n + 1$, and that the statement does not extend to the case where S and T are infinite.