Exercise 18, § 5

In a topological vector space E, let $A_i$ ($1 \leq i \leq n$) be $n$ open non-empty convex sets. a) Show that if the union of the $A_i$ is distinct from E, then every point $x \in E$ not belonging to any of the $A_i$, belongs to a closed linear variety of codimension $n$, that contains $x$ and does not meet any of the $A_i$ (argue by induction on $n$). b) If the intersection of the $A_i$ is empty, show that there exists, in E, a closed linear variety of codimension $n - 1$ that does not meet any of the $A_i$ (same method).