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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.

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    http://math.stackexchange.com/questions/56747/fodors-lemma-on-singular-cardinals – 2017-01-13

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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)