I'm reading the Baumgartner and Dordal paper "Adjoining dominating functions" and I have a problem with the Theorem 4.1 proof.
My problem is the following: Suppose that $P$ is a c.c.c-forcing notion and $P_\lambda$ is the forcing resulting of iterate, with finite support, the forcing $P$ $\lambda$-times (here $\lambda$ is a regular cardinal bigger than $\omega_1$). Let $\{a_\alpha:\alpha<\lambda\}$ be a descending tower in $V^{P_\lambda}$. Consider the set $C:=\{\gamma<\lambda:\{a_\alpha:\alpha<\gamma\}\in V^{P_\gamma}\}$. The authors says that with a standar argument we can find a club set inside of $C$. My question is, which is such "standar argument"? Can someone tell me the way to find such club set?