I try to prove the equivalence between "C is closed in an ordinal $\alpha$" and "each strictly increasing sequence of elements of C of length $< cf\alpha$ converge in C".
For $\Rightarrow$, it's ok.
For $\Leftarrow$ : let $\gamma<\alpha$ ordinal limit such that $Sup(C\cap\gamma)=\gamma$. I want to show that $\gamma\in C$. Let $\beta<\gamma$. There exists $c_0\in C$ such that $\beta<c_0<\gamma$ so $c_0\in C\cap\gamma$. Suppose $c_\xi\in C\cap\gamma$ constructed and define $c_{\xi+1}$ as we done with $c_0$. My questions are :
1) what is the length $\gamma'$ of this sequence ?
2)Do we have $\gamma'<cf\alpha$ ?
3) Does each strictly increasing sequence of ordinals $<\alpha$ is of length $< cf\alpha$ ?
Thanks.
The length of the sequence $c_\xi$ is at least as the cofinality of $\gamma$. Note that $\gamma$ is pretty much any limit ordinal, so it may be any cofinality. Even countable.
Consider the ordinal $\alpha=\omega_2+\omega_1$, and a closed $C$ in $\alpha$ which reflects to $\omega_2$ as a club, namely $C\cap\omega_2$ is closed and unbounded in $\omega_2$, it follows that $\omega_2\in C$, but the cofinality of $\alpha$ is only $\omega_1$.
Of course not, in the case above consider the increasing sequence $\gamma<\omega_2$ below $\alpha$, whose cofinality is $<\omega_2$.