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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.

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  • $\begingroup$ 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
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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 $\alpha=\sup\alpha_n$?

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