I've been trying to prove a property that apparently all functions $g: \omega_1 \rightarrow \omega_1$ have, where $\omega_1$ is the least uncountable ordinal. For $\alpha \in \omega_1$, define $g^\rightarrow(\alpha) = \{g(\beta) : \beta \in \alpha\}$; the claim is then that for all functions $g: \omega_1 \rightarrow \omega_1$, there exists $0< \alpha \in \omega_1$ such that $g^\rightarrow(\alpha) \subseteq \alpha$.
This fact seems strange, and I just don't see why it should even be true. Any thoughts would be appreciated!