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 $A \cap B$?