A short proof of 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 ω^ω‐bounding property. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)