# Introductory Example for Fixpoints

Let be the interval of real numbers. Recall that, by definition of real numbers and complete lattice, is a complete lattice with least lattice element and greatest lattice element . Here is the least upper bound operator on sets of real numbers, also called *supremum* and denoted *sup* in real analysis.

Let function be given by

(It may help you to try to draw .)

#### Part a)

Prove that is monotonic and injective (so it is strictly monotonic).

#### Part b)

Compute the set of fixpoints of .

#### Part c)

Define . (This is in fact equal to when is a monotonic bounded function.)

Compute (prove that the computed value is correct by definition of , that is, that the value is indeed of the set of values). Is a fixpoint of ? Is a fixpoint of ?