Exercise 5, § 3

Let A be a non-empty subset of a vector space E over $\mathbf{R}$ and let p be a sublinear function on E. Suppose that, for every pair of points x, y of A, there exists $z \in A$ such that $p(z - (x + y)) \leq 0$ and that $p(x) \geq 0$ for all $x \in A$. Show that there exists a linear form h on E such that $h(x) \leq p(x)$ in E and that $h(x) \geq 0$ for $x \in A$. (Apply exerc. 4, c) to the union of the $\frac{1}{n} A$ for n (integer) $\geq 1$.)