Proposition 9.1.7.13. Let $\kappa $, $\lambda $, and $\mu $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda $ and $\lambda \trianglelefteq \mu $. Then $\kappa \trianglelefteq \mu $.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $S$ be a $\mu $-small set. Using our assumption that $\lambda \trianglelefteq \mu $, we can choose a $\mu $-small collection $\{ S_ i \} _{i \in I}$ of $\lambda $-small subsets of $S$ such that every $\lambda $-small subset of $S$ is contained in some $S_ i$. For each $i \in I$, our assumption that $\kappa \trianglelefteq \lambda $ guarantees that we can choose a $\lambda $-small collection $\{ S_{i,j} \} _{j \in J_{i} }$ of $\kappa $-small subsets of $S_{i}$ such that every $\kappa $-small subset of $S_{i}$ is contained in some $S_{i,j}$. Then $\{ S_{i,j} \} _{i \in I, j \in J_ i}$ is a $\mu $-small collection of $\kappa $-small subsets of $S$, and every $\kappa $-small subset of $S$ is contained in some $S_{i,j}$. $\square$