Take the tour ×  I found in Brezis' Analyse Fonctionelle the
Hahn-Banach theorem ("second geometric form") and there is a
passage I can't understand. In newest versions of this book the
proof has been modified.  Theorem. Let $A \subset E$ and $B
\subset E$ be two nonempty convex subsets such that $A \cap B =
\emptyset$. Assume that $A$ is closed and $B$ is compact. Then
there exists a closed hyperplane that strictly separates $A$ and
$B$.  Proof. For $\epsilon > 0$ let $A_\epsilon =
A+B(0,\epsilon)$ and $B_\epsilon = B+B(0,\epsilon)$; those sets
are convex and nonempty. They are disjoint too (otherwise, we
can find a sequence $\epsilon_n \to 0$ and $x_n \in A$, $y_n \in
B$ such that $\|x_n - y_n \|< 2\epsilon_n$; so we can extract a
subsequence $y_n \to y \in A \cap B$). $\quad$ [CUT]  My
question is:  How can we extract such a subsequence when we
don't know if $A_\epsilon \cap B_\epsilon$ has elements in
common with $A \cap B$?  share|improve this question add comment
4 Answers  up vote 5 down vote accepted