Exercise 10, § 5

In the normed space $E = l^1(\mathbf{N})$ of summable sequences of real numbers $x = (\xi_n)_{n \in \mathbf{N}}$, let D be the line defined by the relations $\xi_n = 0$ for $n \geq 1$. Show that there exist two increasing sequences $(\alpha_n), (\beta_n)$ of real numbers > 0 such that the convex set A defined by the inequalities $\xi_0 \geq |\alpha_n \xi_n - \beta_n|$ for $n \geq 1$ is closed, non-bounded does not meet D and that there is no closed hyperplane separating A from D (choose $\alpha_n$ and $\beta_n$ so that $A - D$ is everywhere dense).