Regressive function on on singular cardinals I wonder why, if κ is a singular cardinal with cof(κ) > ω, then it must hold that:
For every regressive function
f∶ κ → κ, there is a stationary subset S of κ such that ran(f↾S) is
bounded below κ.
I was trying to use somehow Fodor's lemma but didn't do much.

Fix $\langle t_\alpha:\alpha<{\rm cf}(\kappa)\rangle$ strictly increasing, continuous, and cofinal in $\kappa$. Let $f\!:\kappa\to\kappa$ be regressive, and define $g\!:{\rm cf}(\kappa) \to{\rm cf}(\kappa)$ by setting $g(\alpha)=\beta$ iff $\beta$ is least such that $f(t_\alpha)<t_{\beta+1}$. Since $g$ is regressive, there is a stationary set $T\subseteq{\rm cf}(\kappa)$ where it is constant, say $g(\alpha)=\beta$ for all $\alpha\in T$. Note that $S=\{t_\alpha:\alpha\in T\}$ is stationary in $\kappa$, and that $f(\gamma)<t_{\beta+1}<\kappa$ for all $\gamma\in S$.
