Reducing ordinals in the representation of a limit ordinal as $\bigcup_{\beta < \alpha} \beta$

Question

Let $\alpha$ be a limit ordinal such that $\kappa$ < $\alpha$ < $\kappa^{+}$ where $\kappa$ is initial ordinal (so $|\alpha| = \kappa$).

I want to know whether or not I can find a sequence {$\beta_{i}$} where $ i < \kappa$ such that $\beta_{i} < \beta_{j} $ iff $i<j$, and $\bigcup_{i < \kappa} \beta_{i} = \alpha$.

Answer 1

Not necessarily: you cannot do it if $\kappa>\omega$ and $\alpha=\kappa+\omega$. For regular $\kappa$, like $\omega_1$, you can do it if and only if the cofinality of $\alpha$ is $\kappa$.