## Chugging along

Keeping the notation the same as in the last post, we begin our quest to prove ${\lambda_1\leq\lambda_2}$. The proof is not surprising. Start by letting

$\displaystyle F\subseteq\prod({\sf Reg}\cap(\kappa,\mu)) \ \ \ \ \ (1)$

be a family of cardinality ${\lambda_2}$ “covering all small products”. We let ${\chi}$ be a sufficiently large regular cardinal, and let ${M}$ be an elementary submodel of ${H(\chi)}$ containing the relevant parameters, with

$\displaystyle F\subseteq M. \ \ \ \ \ (2)$

Our plan is to prove that

$\displaystyle \mathcal{P}:=M\cap [\mu]^\kappa \ \ \ \ \ (3)$

is a family satisfying the needed “covering requirements”. Said another way, we will prove that any element of ${[\mu]^{<\theta}}$ can be covered by a union of fewer than ${\sigma}$ sets from ${\mathcal{P}}$. Since

$\displaystyle |\mathcal{P}|=|M|=\lambda_2, \ \ \ \ \ (4)$

this will establish ${\lambda_1\leq \lambda_2}$.