limit ordinal as a countable sequence

Question

If $a$ is a limit ordinal then is it true that there exist (can it be increasing ?) $a_n < a $ such that $a= \bigcup \{a_n : n \in \omega \}$ ?

Answer 1

No, this only holds if the ordinal has countable cofinality. For example, $\omega_1$, the first uncountable ordinal, has uncountable cofinality (assuming choice). That is, given a sequence $(a_n)_{n < \omega}$ with $a_n < \omega_1$, we have that each $a_n$ is countable and so $\bigcup_{n < \omega} a_n$ is countable. Thus we cannot have $\omega_1 = \bigcup_{n < \omega} a_n$.