On $\kappa$-filtrations and clubs.

Question

Let $A$ be a set of cardinality $\kappa$. A $\kappa$-filtration of $A$ is an indexed sequence $\{A_{\nu}:\nu<\kappa\}$ such that for all $\mu,\nu<\kappa$:

1. the cardinality of $A_{\nu}$ is $<\kappa$;
2. if $\mu<\nu$ then $A_{\mu}\subseteq A_{\nu}$;
3. If $\nu$ is a limit in $\kappa$, then $A_{\nu}=\bigcup_{\mu<\nu} A_{\mu}$;
4. $A=\bigcup_{\nu<\kappa} A_{\nu}$.

My question is why the following (Exercise II.18 in Eklof and Mekler's Almost Free Modules) holds:

If $\{A_{\mu}:\mu<\kappa\}$ is a $\kappa$-filtration of a set $A$ of cardinality $\kappa$, being $\kappa$ regular, there is a club $C$ in $\kappa$ such that for all $\nu\in C$, $|A_{\nu^{+}}\setminus A_{\nu}|=|\nu^{+}\setminus \nu|$, where $\nu^{+}$ denotes $\inf\{\alpha\in C:\alpha>\nu\}$.

A hint would be enough. Thank you.

Answer 1

HINT: Let $C=\{\nu<\kappa:|A_\nu|=|\nu|\}$, and show that $C$ is closed and unbounded.