I am trying to study PCF theory by reading Don Monk's very detailed and helpful notes and the handbook article by Abraham and Magidor. The definitions of the concepts mentioned below are according to the notes and the article, which I believe are standard. Here is the link to the notes: http://euclid.colorado.edu/~monkd/full.pdf.
On page 796 Monk claims to give an example of a sequence of ordinal functions that has no strongly increasing subsequence. In particular, he claims that the example depends on Lemma 33.29, in which the $f=\langle f_\xi : \xi < \kappa^+\rangle$ is constructed. From the context it seems that this is supposed to be an example of $<_I$ increasing sequence without strongly increasing subsequence (since the mere existence of a sequence not having strongly increasing subsequence is trivial), but the $f$ here is merely $<_I$ increasing for the first $\kappa$ entries, so it is not $<_I$ increasing. Also the explanation of why $f$ cannot have a strongly increasing subsequence that Monk gives does not depend on the fact that $f$ is $<_I$ increasing for the first $\kappa$ entries at all.
So my question is what is Monk trying to do here? Or do I misinterpret something?
Thank you very much!