I'm trying to solve exercise 8.20 in Davey & Priestley's "Introduction to Lattices and Order". The problem asks me to prove the third CPO fixpoint theorem: If $P$ is a directed complete partial order and $F : P \to P$ satisfies $\forall p \in P,\; p \leq F(p)$, then $P$ contains a minimal fixed point of $F$.
The problem is stated as: 
I believe that I have solved all four parts of the problem except for one annoying difficulty with part (i) and (ii): I have shown that the set $Z$ is $F$-invariant ($F(Z) \subseteq Z$), and also that every directed subset of $Z$ has a least upper bound in $Z$, but I am having trouble proving that $Z$ is non-empty. That is: I can't figure out how to show that there exists at least one roof of $F$ within the directed CPO $P_0$. Without this we only know that if at least one roof element existed, then we would have $Z_x = P_0$, and also $Z = P_0$.
I am working on this problem for school assignment, so I'll be very happy if someone can offer me a hint.
So far I've managed to recognize that assuming that $Z$ is empty causes $P_0$ to contain infinite strictly descending chains. I've been trying to use this fact to contradict $P_0$ being a directed CPO, but I haven't had any luck so far.
Thanks.