TITLE: Exercise 24.13 of T. Jech's *Set Theory*
QUESTION [8 upvotes]: Having struggled my way through most of chapter 24 of Jech's Set Theory, I'm stuck on the very last part of the very last question, 24.13:

Let $I=I_{NS}$ be the nonstationary ideal on $\omega_1$, let $c_{\gamma}$, $\gamma<\omega_1$, be the constant functions (with value $\gamma$) on $\omega_1$, and let $d(\alpha)=\alpha$ be the diagonal function. The function $d$ is a least upper bound, but not an exact upper bound of the set $\{c_{\gamma}:\ \gamma<\omega_1\}$, in $<_I$.

I have shown that $d$ is a least upper bound, but I am unable to find an $f\in\omega_1^{\omega_1}$ with $f<_Id$ such that $c_{\gamma}<_If$ for all $\gamma<\omega_1$, to show that $d$ is not an exact upper bound. Hints are very welcome.

REPLY [3 votes]: In order to show that $f$ is not an exact upper bound you need to find a $g \lt_I d$ that is not bounded by any $c_\gamma$.
Using Solovay's theorem let $\{S_\alpha:\alpha \lt \omega_1\}$ be a partition of $\omega_1$ to $\omega_1$ disjoint stationary sets. Then define $g(\alpha) = min \{i \le \alpha : \alpha \in S_i \}$ if it exists, 0 otherwise.