Let $pcf(a)$ denote the set of regular cardinals such that $J_{\leq \lambda} - J_{<\lambda} \neq \emptyset$ and let $maxpcf(a)$ denote the maximum of $pcf(a)$. The $J_{\leq \lambda}$ are the usual ideals built up inductively in which you throw in all set on which you can have a scale mod $J_{<\lambda}$

There is a fact I want to prove: Let $a$ be a set of regular cardinals such that $a < min(a)$. Then there exists a family $\Im \subseteq \prod a$ with |$\Im$|=$maxpcf(a)$ such that $\forall g\in \prod a$ $\exists f\in \Im$ with $g < f$ (pointwisely). we say that such a family dominates.

I know the proof which uses generators of the $J_{<\lambda}$ ideals, however, since proving that such generators exists is another thing itself, I would like to be able to prove the fact without using generators. I know there is a proof using Hull and Substructures arguments but it is the first time I am using these objects.

Can someone help me please finish the proof? Here is the beginning:

We do an induction on $maxpcf(a)$. When we have $b\subseteq a$ with $maxpcf(b) < maxpcf(a)$ assume for the induction that there exists a family $\Im_b \subseteq \prod b$ with $g< f$.

Let $\delta$ be strictly bigger than $maxpcf(a)$. We need to constuct a continuous chain of elementary substructures $M_{\alpha}$ of $V_\delta$, for $\alpha < |a|^+$, such that |$M_\alpha$|$\leq maxpcf(a)$ and $\forall \alpha<|a|^+, <{M_\beta :\beta < \alpha}> \in M_{\alpha+1}$. If \alpha is a limit then we just take unions: $M_{\alpha+1}= \bigcup_{\beta < \alpha+1} M_\beta$.

If $M_\alpha \bigcap \prod a$ is dominating, we are done. If not then $\exists g_\alpha \in \prod a$ such that $\neg g_\alpha < f, \forall f \in M_\alpha \bigcap \prod a$. One can assume that for $\alpha'<\alpha$ then $g_\alpha' < g_\alpha$ everywhere. This entails that $g_\alpha \in M_{\alpha+1}$.

Now I am stuck here, I want to arrive at a contradiction using my induction hypothesis.