Let $\Lambda$ be the set of all countable limit ordinals. Does there exist an injective function $f:\Lambda\to\omega\_1$ with the properties:


1. $\forall \lambda\in\Lambda:~f(\lambda)<\lambda$
2. $\forall\alpha<\omega\_1~~\exists\beta<\omega\_1~~\forall\lambda>\beta:~f(\lambda)>\alpha$ ?



Function on the set of limit countable ordinals
No. The first property is known as $f$ being *regressive*. [Fodor’s Lemma](https://en.m.wikipedia.org/wiki/Fodor%27s_lemma) says that any regressive function on a stationary set is constant on a stationary subset. In particular, because $\Lambda$ is club (and thus stationary), such $f$ cannot be injective.


