Exercise 3, § 6

In $\mathbf{R}^n$ let $(D_i)_{1 \leq i \leq m}$ be a family of $m \geq 2n$ closed half-spaces determined by hyperplanes passing through 0. Show that if, for any $2n$ of these half spaces, there exists a point $\neq 0$ in their intersection, then there exists a point $\neq 0$ in the intersection of all the $D_i (1 \leq i \leq m)$ (cf. II, p. 66, exerc. 10).