The set of ordinals $< \alpha$ of a given cofinality $\kappa< \text{cf}(\alpha)$ is stationary

Question

I don't understand the last line of the following proof :

I understand that we climbed up to $g(\kappa)$ in $\kappa$ steps (i.e. cf$\big(g(\kappa)\big) \leq \kappa$) but can we be sure that we did in as quickly as one possibly can ? Put differently how can we but sure that $g(\kappa) \in S$ ?

Source : http://euclid.colorado.edu/~monkd/m6730/gradsets09.pdf

Answer 1

Note that $g$ is a strictly increasing continuous function, and $\kappa$ is regular. So this is reduced to showing that $g(\kappa)$ has cofinality $\kappa$.

One direction is trivial, it is certainly at most $\kappa$, but on the other hand, if the cofinality was $\mu<\kappa$, by considering the preimages of that $\mu$-sequence we would obtain a $\mu$-cofinal sequence in $\kappa$.