Let $A=\{ \alpha_n | n \in \omega \}$ a monotonic increasing sequence. Can we say that there exists an $\alpha \in ON$ such that $\alpha = \lim_{n \in \omega} \alpha_n$?
i am asking this because I am trying to show that the set of fixed points of a normal function is a club set/ I know that there is a proof in the literature but I want my own proof. Proving the "closed" part is easy. For the "unbounded" part. I suppose it is bounded by $\alpha$ and then apply f (the normal function). So $\beta_0 = \alpha$ $\beta_n = f^n(\alpha)$. The only thing left is to say that $\{ \beta_n \}$ has a limit in ON..
tnx
Hint: if increasing (@DanielFischer), consider $\alpha=\bigcup_{n\in\omega}\alpha_n$.