There are two versions of the Proper Iteration Lemma. The stronger version can be used to give simpler proofs of iteration theorems . In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ωω-bounding property.

There are two versions of the Proper Iteration Lemma. The stronger (but less well‐known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ωω‐bounding property. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).