For any limit ordinal $\alpha$, $C$ is a c.u.b. set (closed unbounded set) in $\alpha$ iff $C$ is closed and unbounded in $\alpha$.
In Kunen's book, the Lemma of II 6.8 says:If $cf(\alpha) > \omega,$ then the intersection of any family of less than $cf(\alpha)$ c.u.b. subsets of $\alpha$ is c.u.b.
I think the his proof is a little complex. I prove it by this.
Prove: Let $C_\beta$ be c.u.b. in $\alpha$ for $\beta< \lambda$, where $\lambda
However, I proof the lemma without applying the condition $cf(\alpha) > \omega$. I don't know where I am wrong. Could someone give some suggestions to me on this question? Thanks:)
In Kunen's argument, why does the $g^\omega(\xi)$ is in every $C_\beta$? It is said" for each $\beta$, $C_\beta$ is unbounded in $g^\omega(\xi)$, so $g^\omega(\xi)\in C_\beta$." I can't reach this point. Any help will be appreciated:P