Iterated forcing and CH 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.
 A: Here is a sketch.
We may assume that each $\dot Q_\alpha$ has $\omega_1$ as its universe; in which case the underlying set of $P_{\alpha+1}$ can be taken to be $P_\alpha\times\omega_1$ and the ordering defined as usual. This shows that, by induction each $P_n$ has cardinality $\aleph_1$ (at most) and, by CH, so does $P_\omega$. As $P_\omega$ is proper for every name $\dot x$ of a real and every $p\in P_\omega$ there are $q\le p$ and a countable subset $\dot y$ of $P_\omega\times\omega$ (which then acts as a name of a subset of $\omega$) such that $q$ forces $\dot x=\dot y$. This means that those countable names produces the power set of $\omega$, so that, by CH again, in $V^{P_\omega}$ there are $\aleph_1$ many subsets of $\omega$.   
A: 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.  
