Understanding preservation theorems: chapter VI of Proper and Improper Forcing,I

We present an exposition of Section VI.1 and most of Section VI.2 from Shelah’s book Proper and Improper Forcing. These sections offer proofs of the preservation under countable support iteration of proper forcing of various properties, including proofs that ${omega^omega}$ -bounding, the Sacks property, the Laver property, and the P-point property are preserved by countable support iteration of proper forcing.