I'm asking a question following the reading of Set Theory from Jech (page 126).

For the record, a $\kappa$-sequence of functions $\langle f_\alpha : \alpha < \kappa\rangle$ is called a $\kappa$-scale if $f_\beta < f_\alpha$ whenever $\beta < \alpha$, and if for every $g\colon \omega \rightarrow \omega$ there exists an $\alpha$ such that $g < f_\alpha$.

Suppose that $\aleph_1 = 2^{\aleph_0}$. A scale $\langle f_\alpha : \alpha < \omega_1 \rangle$ is constructed by transfinite induction to $\omega_1$:

Let $\{g_\alpha : \alpha < \omega_1\}$ enumerate all functions from $\omega$ to $\omega$. At stage $\alpha$, we construct, by diagonalization, a function $f_\alpha$ such that for all $\beta < \alpha$, $f_\alpha > f_\beta$ and $f_\alpha > g_\beta$. Then $\langle f_\alpha : \alpha < \omega_1 \rangle$ is an $\omega_1$-scale.

**I don't see how to manage the transfinite induction in order to avoid for limit ordinals that $f_\alpha$ becomes infinite.**

Thanks for your support!