Final answer:
The supremum of the set of all cluster points of a finite bounded set cannot exceed the supremum of the set itself. The cluster points are the points of the set or none; hence, sup A^ is either equal to or less than sup A.
Step-by-step explanation:
The question is asking to compare the supremum of a finite bounded set A with the supremum of the set of all its cluster points, denoted as A^. A cluster point of a set is a point such that every neighborhood around that point contains at least one point from the set that is different from the point itself. In the case of a finite set, the cluster points are actually the points of the set itself because there are no other points arbitrarily close to any point that are also members of the set.
Now, since the cluster points of A are the points of A itself, the set A^ includes elements from A or none at all if A has no cluster points. As such, the supremum of the set of all cluster points (sup A^) cannot be greater than the supremum of set A (sup A), because it is either equal to sup A or there is no supremum for an empty set of cluster points. Therefore, sup A^ ≤ sup A.