TITLE: Iterated forcing and CH
QUESTION [8 upvotes]: I need some help with this theorem: if $P_\beta=\langle P_\alpha,\dot{Q}_\alpha:\alpha\leq\beta\rangle$, $\beta<\omega_2$, is a CSI of proper forcings, $P_\alpha\Vdash \lvert \dot{Q}_\alpha\rvert\leq\aleph_1$, and CH holds in the ground model, then $P_\beta$ forces the CH in the generic extension.
A proof of this fact appears in the Handbook, but I need a diferent one. Just for simplicity, suppose $\beta=\omega$, then the problem is how to prove that $P_\omega$ forces the CH. Can someone give a clue or an idea of how to proceed?
Thanks.

REPLY [3 votes]: For any $\alpha < \omega_2$, the forcing notion $P_\alpha$ has a dense subset $P'_\alpha$ of cardinality $\aleph_1$.  
This is proved in Shelah's "Proper Forcing" book. Jakob Kellner and I also give a sketch of this proof in our review of Uri Abraham's survey article on proper forcing (MR2768684). 
As a consequence, each $P_\alpha$ will satisfy the $\aleph_2$-cc. The rest of the proof is as in the handbook (I do not know any essentially different proof): Counting "nice names" (i.e., names of reals that are defined from countably many antichains) shows that CH is preserved.