\DeclareMathOperator{\pp}{pp}

\def\pcf{\rm{pcf}}

\DeclareMathOperator{\cov}{cov}

\def\cf{\rm{cf}}

\def\REG{\sf {REG}}

\def\restr{\upharpoonright}

\def\bd{\rm{bd}}

\def\subs{\subseteq}

\def\cof{\rm{cof}}

\def\ran{\rm{ran}}

\DeclareMathOperator{\ch}{Ch}

\DeclareMathOperator{\PP}{pp}

\DeclareMathOperator{\Sk}{Sk}

\)

We will follow Abraham-Magidor for a little bit.

Assume the following:

- $A$ is a progressive set of regular cardinals
- $\kappa$ is a regular cardinal with $|A|<\kappa<\min(A)$,
- $\langle f_\xi:\xi<\delta\rangle$ is a sequence of functions in $\prod A$ with $\cf(\delta)=\kappa$.

Given a club $E\subseteq\delta$ of order-type $\kappa$, we define

$h_E = \sup\{f_\xi:\xi\in E\}$.

Note the following:

- $h_E$ is in $\prod A$ as $\kappa<\min(A)$,
- if $C\subseteq E$ are clubs in $\delta$, then $h_C\leq h_E$ in $\prod A$, and
- there is a club $C\subseteq \delta$ of order-type $\kappa$ such that $h_C\leq h_E$ for any club $E\subseteq \delta$.

The last point follows by an easy proof by contradiction. If there is no such club $C$, then we can build a decreasing sequence $\langle E_\alpha:\alpha<|A|^+\rangle$ of clubs in $\delta$ such that

- $E_\alpha=\bigcap_{\beta<\alpha}E_\beta$ if $\alpha$ is a limit ordinal, and
- $h_{E_{\alpha}}\nleq h_{E_{\alpha+1}}$

For each $\alpha<|A|^+$, we can find $a_\alpha\in A$ such that

$h_{E_{\alpha+1}}(a_\alpha)< h_{E_\alpha}(a_\alpha)$

and then stabilize to find a single $a\in A$ such that

$h_{E_{\alpha+1}}(a)< h_{E_{\alpha}}(a)$

for unboundedly many indices $\alpha$.

Since the sequence of clubs $E_\alpha$ is decreasing, we get an infinite decreasing sequence of ordinals. Contradiction.

Now suppose $J$ is an ideal over $A$, and the sequence $\langle f_\xi:\xi<\delta\rangle$ is $<_J$-increasing. In this case, if $C$ is the "minimal" club as above, then

$\xi\in C\Longrightarrow f_\xi\leq f_C$,

and hence $f_C$ is a $<_J$-upper bound for the sequence $\langle f_\xi:\xi<\delta\rangle$.

The function $f_C$ is called the

*minimal club-obedient bound of*$\langle f_\xi:\xi<\delta\rangle$
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