Recall that by $\lambda(\omega^{\omega^\kappa+\xi_1},n) = \omega^{\omega^\kappa}\theta(\xi_1,n)$. For each $n$, define $\eta(\gamma,n) = \omega^{\xi_l}m_l+\omega^{\xi_{l-1}}m_{l-1}+ \cdots +\omega^{\xi_1}(m_1-1) + \theta(\xi_1,n)$. We will show that $(\eta(\gamma,n))_{n\in{\mathbb{N}}}$ has the desired property. Note that the Cantor Normal Form of $\beta\gamma$ is $\beta\gamma = \omega^{\omega^\kappa+\xi_l}m_l+\omega^{\omega^\kappa+\xi_{l-1}}m_{l-1}+ \cdots +\omega^{\omega^\kappa+\xi_1}m_1$. Hence, by : $$\begin{aligned}
\lambda(\beta\gamma,n) &=&  \omega^{\omega^\kappa+\xi_l}m_l+\omega^{\omega^\kappa+\xi_{l-1}}m_{l-1}+ \ldots +\omega^{\omega^\kappa+\xi_1}(m_1-1) + \lambda(\omega^{\omega^\kappa+\xi_1},n)\\
&=& \omega^{\omega^\kappa+\xi_l}m_l+\omega^{\omega^\kappa+\xi_{l-1}}m_{l-1}+ \ldots +\omega^{\omega^\kappa+\xi_1}(m_1-1) + \omega^{\omega^\kappa}\theta(\xi_1,n)\\
&=& \omega^{\omega^\kappa}(\omega^{\xi_l}m_l+\omega^{\xi_{l-1}}m_{l-1}+ \ldots +\omega^{\xi_1}(m_1-1) + \theta(\xi_1,n))
 = \beta\eta(\gamma,n).\end{aligned}$$

The proof of Proposition \[P:2.6\], in particular the definition of $(\eta(\gamma,n))_n$, implies the following. Let $\xi$ be a countable ordinal number and $\gamma \le \beta = \omega^{\omega^\xi}$ be a limit ordinal number. If $\gamma = \omega^{\xi_1}$, then $$\label{E:2.15}
\eta(\gamma,n) = \theta(\xi_1,n),\text{ for all }n\in{\mathbb{N}}.$$ Otherwise, if the Cantor normal form of $\gamma$ is $$\gamma=\omega^{\xi_l}m_l+\omega^{\xi_{l-1}}m_{l-1}+ \ldots +\omega^{\xi_1}m_1$$ and $\gamma_1 =\omega^{\xi_l}m_l+\omega^{\xi_{l-1}}m_{l-1}+ \ldots \omega^{\xi_j}m^{(1)}_j$, $\gamma_2= \omega^{\xi_j}m_j^{(2)}+\omega^{\xi_{j-1}}m_{j-1}+ \ldots \omega^{\xi_1}m_1$, with $m^{(1)}_j,m^{(2)}_j\in{\mathbb{N}}\cup\{0\}$, $m_j=m^{(1)}_j+m_j^{(2)}$, then we have $$\label{E:2.16}
\eta(\gamma,n) = \gamma_1 + \eta(\gamma_2,n),\text{ for all }n\in{\mathbb{N}}.$$

\[C:2.5\] If $\alpha<\omega_1$ is a limit ordinal it follows that $$\label{E:2.5.1}   {\mathcal S}_\alpha=\big\{ A\in[{\mathbb{N}}]^{<\omega}\setminus\{\emptyset\}: A\in {\mathcal S}_{\lambda(\alpha,\min(A))}\big\}\cup\{\emptyset\}.$$

If we had defined ${\mathcal S}_\alpha$ by , for limit ordinals $\alpha{\!<\!}\omega_1$ where $\big(\lambda(\alpha,n):n{\!\in\!}{\mathbb{N}})$ is any sequence increasing to $\alpha$, then we would not have ensured that the family ${\mathcal S}_\alpha$ is a regular family.

The fine Schreier 