families {#SS:2.3}
--------------------------

We will now define the [*Fine Schreier Sets*]{}. For that we will also need to choose appropriate approximating sequences for limit ordinals. We will define them as a doubly indexed family ${\mathcal F}_{\beta, \alpha}\subset [{\mathbb{N}}]^{<\omega}$, $\alpha\le \beta<\omega_1$. Later in the proof of Theorems A and C, we will fix $\beta$, depending on the Banach space $X$ we are considering.

\[D:2.9\] For a countable ordinal number $\xi$ and $\beta = \omega^{\omega^{\xi}}$, we recursively define an hierarchy of families of finite subsets of the natural numbers $({\mathcal F}_{\beta,\gamma})_{\gamma\le\beta}$ as follows:

-   ${\mathcal F}_{\beta,0} = \{\emptyset\}$,

-   if $\gamma<\beta$ then ${\mathcal F}_{\beta,\gamma+1} = \{\{n\}\cup F: F\in{\mathcal F}_{\beta,\gamma},n{\!\in\!}{\mathbb{N}}\}$ (i.e., ${\mathcal F}_{\beta,\gamma+1} ={\mathcal F}_{\beta,1}\sqcup_<{\mathcal F}_{\beta,\gamma} $), and

-   if $\gamma\le\beta$ is a limit ordinal number, then ${\mathcal F}_{\beta,\gamma} = \cup_{n\in{\mathbb{N}}}({\mathcal F}_{\beta,\eta(\gamma,n)}\cap[n,\infty)^{<\omega})$, where $(\eta(\gamma,n))_n$ is the sequence provided by Proposition \[P:2.6\] (and depends on $\beta$).

It can be easily shown by transfinite induction that each family ${\mathcal F}_{\beta,\gamma}$ is regular. In the literature fine Schreier families are usually defined recursively as a singly indexed family $({\mathcal F}_\alpha)_{\alpha<\omega_1}$ of subsets of $[{\mathbb{N}}]^{<\omega}$. In that case ${\mathcal F}_{0}$ and ${\mathcal F}_\alpha$ are defined for successor ordinals as in (i) and (ii). And if $\alpha$ is 