# Function on the set of limit countable ordinals

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. $$\forall\alpha<\omega_1~~\exists\beta<\omega_1~~\forall\lambda>\beta:~f(\lambda)>\alpha$$ ?
• Though the answer below answers your question, you might be interested to know that an injective function $f:\Lambda\to\omega_1$ automatically satisfies $2$: for any $\gamma\leq\alpha$ there is at most one ordinal $a_\gamma$ such that $f(a_\gamma)=\gamma$. Then $\beta=\sup_{\gamma\leq\alpha}a_\gamma$ will work. – Wojowu Apr 24 '19 at 8:25

No. The first property is known as $$f$$ being regressive. Fodor’s 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.