Exercise 2, § 5

Let $N$ be a hyperplane in a vector subspace $M$ of a vector space $E$ and let $A$ be a convex set in $E$, such that all the points of $A \cap M$ are on the same side of $N$ and which also possesses the following property; for any $y \neq 0$ in $E$, there exists $x \in A \cap M$ such that $x + \lambda y \in A$ for all $\lambda$ such that $|\lambda|$ is sufficiently small. Show that there exists then, a hyperplane $H$ of $E$ such that all the points of $A$ are on the same side of $H$ and such that $H \cap M = N$. (Reduce to the case $N = \{0\}$; if $a \neq 0$ belongs to $A \cap M$, consider the set $\mathscr{U}$ of pointed convex cones with vertex 0 containing $A$ and not containing $-a$; show that there exists a maximal element $C$ of $\mathscr{U}$ and that $C$ is a maximal pointed convex cone (exerc. 1)). Deduce a new proof of the Hahn-Banach theorem.