To prove corollary 2.4.1, let f: X → X be an increasing function on a complete lattice (X, ≿), and let E be the set of fixed points of f. For any S ⊆ E define
S* = {x ∊ X : x ≿ s for every s ∊ S}
S* is the set of all upper bounds of S in X. Show that
1. S* is a complete sublattice.
2. f (S*) ⊆ S*.
3. Let g be the restriction of f to the sublattice S*. g has a least fixed point .
4.  is the least upper bound of S in E.
5. E is a complete lattice.