Regular cardinals and unions

Question

If a cardinal $\kappa$ is regular then it cannot be written as a union of fewer than $\kappa$ sets, each of size less than $\kappa$.

This seems to be a very useful characterization. I have seen a proof or two, but can't grasp all the details... I am horrible with ordinal and cardinal arithmetic. Could someone please give an elementary (as much as possible) proof of this theorem for me?

Answer 1

The simplest proof I can think of is as follows.

Suppose that $\kappa$ is a regular cardinal, let $\lambda < \kappa$ and let $\{ A_\xi : \xi < \lambda \}$ be a family of subsets of $\kappa$ each of cardinality $< \kappa$. As $\kappa$ is regular, then no $A_\xi$ is cofinal in $\kappa$ (since $|A_\xi| < \kappa = \mathrm{cf} ( \kappa)$), meaning that for each $\xi < \lambda$ there is an $\alpha_\xi < \kappa$ such that $A_\xi \subseteq \alpha_\xi$ — what I really mean here is $\beta < \alpha_\xi$ for all $\beta \in A_\xi$. Again by the regularity of $\kappa$ the family $\{ \alpha _\xi : \xi < \lambda \}$ cannot be cofinal in $\kappa$, and so there is an $\alpha < \kappa$ such that $\alpha_\xi < \alpha$ for all $\xi < \lambda$. But then it easily follows that $\bigcup_{\xi < \lambda} A_\xi \subseteq \alpha \subsetneq \kappa$.

There is no cardinal/ordinal arithmetic needed here; all we have used is the definition of cofinality. If there are particular points that are troubling you, please point them out and I can (try to) elucidate further.