In the Jech's textbook proof of Silvers Theorem, specifically in the Lemma 8.15, there are an assumption in the beginning of the proof.
First, the Lemma 8.15 says that, under the assumption that $\aleph_{\alpha}^{\aleph_1}<\aleph_{\omega_1}$ for all $\alpha < \omega_1$, if F is a almost disjoint family of functions contained in a product of the form $\prod_{\alpha<\omega_1}A_\alpha$ such that the set $\{\alpha<\omega_1:|A_\alpha|\leqslant \aleph_{\alpha+1}\}$ is stationary on $\omega_1$, then $|F|\leqslant\aleph_{\omega_1+1}$.
(A family of functions $F$ is almost disjoint if for all $f,g\in F$ such that $f\neq g$, there exist $\beta<\omega_1$ such that for all $\gamma > \beta$ we have $f(\gamma)\neq g(\gamma)$).
In the proof of that lemma, we suppose the lemma 8.16, but before that, there are an assumption: all $A_\alpha$ in the product given above have size at most $\omega_{\alpha+1}$. This is clear for all $\alpha$ in the stationary set given in the hypothesis. Why we can make the assumption for all $\alpha$?
Thank you for the further answers.