A simple proof that any additive basis has only finitely many essential subsets

Let $A$ be an additive basis. We call essential subset'' of $A$ any finite subset $P$ of $A$ such that $A \setminus P$ is not an additive basis and that $P$ is minimal (for the inclusion order) to have this property. A recent theorem due to B. Deschamps and the author states that any additive basis has only finitely many essential subsets (see Essentialit'e dans les bases additives, J. Number Theory, 123 (2007), p. 170-192’'). The aim of this note is to give a simple proof of this theorem.