Exercises for Week 7

Let $\lambda$ be a limit ordinal. We define the cofinality of $\lambda$, written $\text{cf}(\lambda)$, to be the minimum order type of $(A;<)$ for all subsets $A \subseteq \lambda$ with $\sup A = \lambda$. We say that $\lambda$ is regular if $\text{cf}(\lambda) = \lambda$.

1. Prove that $\omega \le \text{cf}(\lambda)$.
2. Prove that $\text{cf}(\lambda) \le |\lambda|$.
3. Prove that if $\lambda$ is regular, then $\lambda$ is a cardinal. (So we call a regular limit ordinal $\lambda$ a "regular cardinal" instead of a "regular ordinal".)
4. Prove that $\text{cf}(\lambda)$ is a regular cardinal. Hint: you can do this directly from the definitions, but it might help to think in terms of cofinal increasing functions instead.