a limit ordinal is defined as in (iii) where we do not assume that the approximating sequence $(\eta(\alpha,n))_{n\in{\mathbb{N}}}$ does not depend of any $\beta\ge \alpha$.

Let $\xi$ be a countable ordinal number and $\xi_1\le\omega^\xi$. If and $\beta = \omega^{\omega^\xi}$ and $\gamma = \omega^{\xi_1}$, it follows by that $\eta(\gamma,n)=\theta(\gamma,n)$ for $n\in{\mathbb{N}}$. The choice of $(\theta(\xi_1,n))_{n\in{\mathbb{N}}}$ may be done so that alongside , we also have $$\label{E:2.17}
{\mathcal F}_{\beta,\eta(\gamma,n)} = {\mathcal F}_{\beta,\theta(\xi_1,n)} \subset {\mathcal F}_{\beta,\theta(\xi_1,n+1)} = {\mathcal F}_{\beta,\eta(\gamma,n+1)}.$$ This can be achieved by possibly adding to $\theta'(\xi_1,n)$ a large enough natural number.

\[P:2.10\] Let $\xi$ be a countable ordinal number and $\beta = \omega^{\omega^\xi}$. Assume that for all limit ordinals limit ordinals $\gamma\le\beta$ the approximating sequence $(\eta(\gamma,n))_n$ satisfies conditions and , and for the case $\gamma = \omega^{\xi_1}$ the approximating sequence $(\theta(\xi_1,n))_n$ satisfies condition . Then, for all $\gamma\le\beta$, whose Cantor normal form is $$\gamma=\omega^{\xi_l}m_l+\omega^{\xi_{l-1}}m_{l-1}+ \cdots +\omega^{\xi_1}m_1$$ we have that that $$\label{E:2.11.1} {\mathcal F}_{\beta,\eta(\gamma,n)}\subset  {\mathcal F}_{\beta,\eta(\gamma,n+1)} \text{ for all $n\in{\mathbb{N}}$, if $\gamma$ is a limit ordinal,}$$ and if for some for some $1\le j\le l$, $\gamma_1 =\omega^{\xi_l}m_l+\omega^{\xi_{l-1}}m_{l-1}+ \cdots +\omega^{\xi_j}m^{(1)}_j$ and $\gamma_2= \omega^{\xi_j}m_j^{(2)}+\omega^{\xi_{j-1}}m_{j-1}+ \cdots +\omega^{\xi_1}m_1$ with $m^{(1)}_j,m^{(2)}_j\in{\mathbb{N}}\cup\{0\}$, $m_j=m^{(1)}_j+m_j^{(2)}$, then $$\label{E:2.11.2} {\mathcal F}_{\beta,\gamma} = {\mathcal F}_{\beta,\gamma_2}\sqcup_<{\mathcal F}_{\beta,\gamma_1}.$$

We will show and simultaneously by transfinite 