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 1) what is the length $\gamma'$ of this sequence ? 2)Do we have $\gamma' 3) Does each strictly increasing sequence of ordinals $<\alpha$ is of length $< cf\alpha$ ? Thanks.