
fixed point theorem
Prove that for every function $\displaystyle f: \omega_1 \rightarrow \omega_1$, there is a $\displaystyle \beta \in \omega_1$ such that $\displaystyle f[\beta] \subseteq \beta$.
Hint: A fixed point theorem may be useful. We may us the Countable Principle of Choice, but not the Axiom of Choice.

Are you referring to Knaster–Tarski fixpoint theorem? Since it requires a monotonic function, maybe one can consider $\displaystyle g(\beta)=\bigcup_{\alpha\le\beta}f(\alpha)$. Then $\displaystyle g$ is monotonic, so it has a fixpoint $\displaystyle \beta_0$: $\displaystyle g(\beta_0)=\beta_0$, which implies $\displaystyle f(\beta_0)\subseteq\beta_0$. And the Countable Principle of Choice is used to show that $\displaystyle \omega_1$ is a complete lattice. I am not sure about this, but it may be a start...