 Solution

Fodor lemma says that if $\lambda$ is a regular uncountable cardinal, $S\subseteq \lambda$ is stationary, and $f:S\to \lambda$ is regressive, thenHere a regressive function satisfies $f(\alpha )<\alpha$ at every nonzero $\alpha \in S$; removing $0$ has no effect on stationarity. Both regularity and stationarity are part of the hypotheses.