out that the sequence $(\theta(\xi,n))_n$ also depends on $\alpha$.

\[P:2.6\] Assuming the approximating sequences $\big(\lambda(\alpha,n):n\in{\mathbb{N}}\big)$ satisfy for all limit ordinals $\alpha$ the above conditions. It follows for all $\gamma<\omega_1$, with $$\gamma=\omega^{\xi_l}m_l+\omega^{\xi_{l-1}}m_{l-1}+ \ldots +\omega^{\xi_1}m_1,$$ being the Cantor Normal Form, that $$\begin{aligned}
\label{E:2.6.1}& {\mathcal S}_{\lambda(\gamma,n)}\subset  {\mathcal S}_{\lambda(\gamma,n+1)} \text{ for all $n\in{\mathbb{N}}$, if $\gamma$ is a limit ordinal.}\\
\label{E:2.6.2} &{\mathcal S}_\gamma= {\mathcal S}_{\gamma_2}[{\mathcal S}_{\gamma_1}], \text{ where for some $j=1,2,\ldots ,l$}\\
& \gamma_1 =\omega^{\xi_l}m_l{\!+\!}\omega^{\xi_{l-1}}m_{l-1}{\!+\!}\ldots{\!+\!}\omega^{\xi_j}m^{(1)}_j \text{ and }
\gamma_2= \omega^{\xi_j}m_j^{(2)}{\!+\!}\omega^{\xi_{j-1}}m_{j-1}{\!+\!}\ldots{\!+\!}\omega^{\xi_1}m_1,\notag\\
&\qquad \text{with $m^{(1)}_j,m^{(2)}_j\in{\mathbb{N}}\cup\{0\}$,   $m_j=m^{(1)}_j+m_j^{(2)}$.}\notag
\intertext{and 
if $\beta = \omega^{\omega^\kappa}$ and $\gamma$ is a limit ordinal with $\gamma \le \beta$, then}
&\label{E:2.6.3}\text{there is a sequence $(\eta(\gamma,n))_n$ increasing to $\gamma$
so that
 $\lambda(\beta\gamma,n) = \beta\eta(\gamma,n)$}\end{aligned}$$ (this sequence $(\eta(\gamma,n))_n$ can depend on $\beta$).

We first will prove , simultaneously for all $\gamma<\omega_1$. Assume that our claim is true all $\tilde\gamma<\gamma$. follows from Proposition \[P:2.4\].

If $l=m_1=1$ we deduce from the choice of $\lambda(\gamma,n)$, $n\in{\mathbb{N}}$, in that case (see , and ). If $l\ge 2$ or $m_2\ge 2$ we deduce from and the induction hypothesis that $$\begin{aligned}
{\mathcal S}_{\lambda(\gamma,n)}&={\mathcal S}_{\omega^{\xi_k}m_k+\ldots+ \omega^{\xi_2}{m_2}+ \omega^{\xi_1}{m_1}+\lambda(\omega^{\xi_1},n)}\\
&=
{\mathcal S}_{\lambda(\omega^{\xi_1},n)}[{\mathcal S}_{\omega^{\xi_k}m_k+\ldots+ \omega^{\xi_2}{m_2}+ \omega^{\xi_1}{m_1}}]\\
&\subset
{\mathcal S}_{\lambda(\omega^{\xi_1},n+1)}[{\mathcal S}_{\omega^{\xi_k}m_k+\ldots+ \omega^{\xi_2}{m_2}+ \omega^{\xi_1}{m_1}}]=
{\mathcal S}_{\lambda(\gamma,n+1)},\end{aligned}$$ which verifies also in that case

To verify let $\kappa<\omega_1$ with $\beta=\omega^{\omega^\kappa}\ge \gamma$. 