I have many questions about page 2 of this paper http://www.cs.elte.hu/~kope/ss3.pdf.
First, on the top, I try to prove that, if $cf(\delta)\neq\kappa$, then we can choose the $\alpha, \beta$ in an $A_i^\delta$.
- We can prove that there exists an $i<\kappa$ so that the set $\{\delta<\lambda^+ : A_i^\delta \text{ is cofinal in }\delta\}$ is stationary in $\lambda^+$.
- I know that, for all $i<\kappa$, the set $B_i^\delta=\{\alpha<\delta : \exists\beta<\delta, Z_i\cap\alpha=(X_\beta)_i\}$ is cofinal in $\delta$.
- Intuitively, $cf(\delta)<\kappa$.
If $\delta$ is an ordinal then the cardinal of $\{X\subseteq\delta : X \text{ cofinal in } \delta\}$ is $2^\delta$ ?
but I can't "see" why we can take $\alpha,\beta$ in $A_i^\delta$.
Second, in Lemma 3, do we take a particular bijection $\pi$ ? I want to take the sequence $\langle \pi(X_\beta) : \beta<\lambda^+\rangle$ but I have some difficulty to prove that $Z\cap(\lambda\times\alpha)=X_\beta$. I need $\pi(\alpha)=\lambda\times\alpha$...
Third, in the proof of the claim, I want to read $Y_\eta\cap\alpha=(X_\beta)_\eta$ instead of $Y_\eta=(X_\beta)_\eta$. Is it correct ?