# Thread:

1. ## fixed point theorem

Prove that for every function $f: \omega_1 \rightarrow \omega_1$, there is a $\beta \in \omega_1$ such that $f[\beta] \subseteq \beta$.

Hint: A fixed point theorem may be useful. We may us the Countable Principle of Choice, but not the Axiom of Choice.

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