I'm trying to understand the construction of a club set of $\omega_1$ inside a stationary one in a generic extension. Given a stationary set $S\subset\omega_1$ the forcing which force the existence of a club inside $S$ in a generic extension is $P_S$ where $p\in P_S$ are countable closed subsets of $S$ and $p\leq q$ iff $p\cap \max(q)+1=q$.
It's proved that $C=\bigcup G$ is a closed unbounded set in $M[G]$ but I'm having some difficulties to understand why it's closed. More precisely, I would like to know why if $\alpha$ is a limit point of $C$ then it must be a limit point of a $p\in G$ such that $\max(p)>\alpha$.
Thank you in advance.
Cesare.
Suppose that $\alpha$ is a limit point of $C$; $\{p\in P_S:\max p>\alpha\}$ is dense in $P_S$, so there is a $p\in G$ such that $\max p>\alpha$. But $p=C\cap(\max p+1)\supseteq C\cap\alpha$, and $\alpha$ is a limit point of $C\cap\alpha$, so $\alpha$ is a limit point of $p$ and hence in $p$.