Understanding the Existence and Uniqueness of the Least Upper Bound

When discussing the properties of sets and their elements, it is crucial to understand the concept of an upper bound and its least upper bound. An upper bound of a set signifies a value that is greater than or equal to all the elements within the set. However, not all sets with an upper bound have a least upper bound. This article explores the conditions under which a set possesses a least upper bound and provides counterexamples to illustrate when this is not the case.

Introduction to Upper Bounds

In mathematics, particularly in the realm of real analysis, the concept of upper bounds is fundamental. A set A of real numbers is said to have an upper bound if there exists a real number M such that M ≥ a for all a in A. However, the presence of an upper bound does not necessarily guarantee the existence of a least upper bound. This article aims to clarify this concept through detailed explanations and examples.

The Existence of a Least Upper Bound in Real Numbers

Let us consider a set A with an upper bound M. We aim to construct a new set B which consists of all upper bounds of A that are less than or equal to M. Thus, define B as:

B {x : x is an upper bound of A and x ≤ M}

Since A has an upper bound, B is a non-empty subset of the real numbers. The real numbers are well-ordered, meaning every non-empty subset has a least element. Therefore, B also has a least element, denoted by L. To show that L is the least upper bound of A, we proceed with a proof by contradiction.

Proof by Contradiction

Suppose, for the sake of contradiction, that there exists another upper bound y of A such that y L. By the definition of B, y would be an element of B, which contradicts the assumption that L is the least element of B. Therefore, L must be the least upper bound of A. This demonstrates that if A has an upper bound, it indeed has a least upper bound.

A Counterexample: Rational Numbers and the Square Root of 2

The above argument holds true for the set of real numbers. However, it does not apply when dealing with the set of rational numbers. The rational numbers, denoted by Q, are not as well-behaved as the real numbers in terms of completeness, meaning there exist sets of rational numbers that have upper bounds but do not have a least upper bound in Q.

Counterexample: The Set of Rational Numbers

Consider the set of rational numbers whose square is less than 2. Mathematically, this set is denoted as A {x : x ∈ Q, x2 2}.

This set A has an upper bound, the number 2. However, it does not have a least upper bound in the set of rational numbers. This demonstrates that the set of rational numbers is not complete, as there are sets with upper bounds that do not have a least upper bound within the set itself.

Conclusion

The existence of a least upper bound is a fundamental property of the real number system, which is why they are considered complete. In contrast, the rational numbers, while dense, do not necessarily possess this property. Understanding the distinction between the real and rational numbers, and their respective completeness properties, is crucial for further study in real analysis and number theory.