Let $\Lambda$ be the set of all countable limit ordinals. Does there exist an injective function $f:\Lambda\to\omega_1$ with the properties:

- $\forall \lambda\in\Lambda:~f(\lambda)<\lambda$
- $\forall\alpha<\omega_1~~\exists\beta<\omega_1~~\forall\lambda>\beta:~f(\lambda)>\alpha$ ?