Let $\kappa$ be a regular, uncountable Cardinal and let $f:\kappa\rightarrow\kappa$. I'm trying to show that $\{\alpha<\kappa\mid f''\alpha\subseteq\alpha\}$ is club in $\kappa$. I can see why it's closed, but I'm having a hard time seeing why it would be unbounded. Any help would be appreciated.

  • $\begingroup$ I think you need to assume more properties of the function, like being continuous at limits. $\endgroup$ – William May 13 '16 at 2:17
  • $\begingroup$ @William No, that's not needed. $\endgroup$ – Andrés E. Caicedo May 13 '16 at 2:39

Given any $\gamma$, define by induction the following sequence: $$\alpha_0=\gamma+1,\quad \alpha_{n+1}=\max\{\alpha_n,\sup f''\alpha_n\}. $$

What can you say about $\alpha=\sup\alpha_n$?

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.