Prove that a countable inductive poset is directed complete.
This question is from Notes on Set Theory 2nd edition by Yannis Moskovakis. Here, an inductive poset is a poset such that every chain in the poset has a least upper bound.
I am not sure what I can do here. Let $P$ be a countable, inductive partially ordered set, and let $S$ be a directed set. If $S$ is a chain, then it has a least upper bound so we are done. However $S$ need not be a chain.
As the chapter that has this problem has the following theorem, I may be able to use this but I am not sure how to:
Theorem: every monotone mapping $\pi: P \to P$ on an inductive poset to itself has exactly one strongly least fixed point $x^*$ characterized by two properties $$\pi(x^*)=x^*,$$ $$\forall y\in P[\pi(y) \leq y \Rightarrow x^* \leq y]$$
Actually, the theorem on the chapter requires $\pi$ to be countably countinuous, but such condition is later in the book proven to be superflous.
I may be able to use the "least property" of a function $\pi$ satisfying the conditions stated in the theorem, but I am not sure how I can define such a $\pi$.