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Let $A$ be an additive basis. We call ``essential subset'' of $A$ any finitesubset $P$ 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 recenttheorem due to B. Deschamps and the author states that any additive basis hasonly finitely many essential subsets (see ``Essentialit\'e dans les basesadditives, J. Number Theory, 123 (2007), p. 170-192''). The aim of this note isto give a simple proof of this theorem.

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