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