Exercise 5, § 7

In the Banach space $l^\infty(\mathbf{N})$ of bounded sequences $x = (\xi_n)$ of real numbers, let A be the closed convex set defined by the inequalities $-1/n \leq \xi_n \leq 1$ for $n \geq 1$ and $-1 \leq \xi_0 \leq 1$. Show that A has a non-empty interior, that the origin is a frontier point of A and that the facet of 0 in A is not closed. If we give to A the topology induced by that of the product space $\mathbf{R}^N$, show that A is compact but that the facet of 0 in A is not closed in A.