Let $(A_\nu)_{\nu<\mu}$ be an increasing sequence of free abelian groups such that for any limit ordinal $\lambda<\mu$, $A_\lambda=\bigcup_{\nu<\lambda}A_\nu$. Let $A=\bigcup_{\nu<\mu}A_\nu$. Let $\kappa$ be the cofinality of $\mu$.
Assume that $A_\sigma/A_{\nu+1}$ is free whenever $\nu<\sigma<\mu$.
Let $\{p_\sigma:\sigma<\kappa\}$ be a closed unbounded subset of $\mu$ that is strictly increasing. Assume that for every $\sigma<\kappa$ such that $p_\sigma$ is a limit ordinal, $A_{p_\sigma+1}/A_{p_\sigma}$ is free.
A paper says it is "evident" that $A$ is free and that $A/A_{p_\sigma}$ is free for all $\sigma<\kappa$.
It is not evident to me. Why are these statements true?
Source: the proof of Lemma 2.1 in Paul Eklof, "On the Existence of $\kappa$-Free Abelian Groups," Proceedings of the American Mathematical Society 47 (1975), 65-72.