Question

Using properties \((\mathbf{A})-(\mathbf{H})\) of the real numbers and taking Dedekind's theorem (Exercise 1.1 .8 ) as given, show that every nonempty set \(U\) of real numbers that is bounded above has a supremum. Hivt: Let \(T\) be the set of upper bounds of \(U\) and \(S\) be the set of real numbers that are not upper bounds of \(U .\)

Short answer

#Answer# The supremum of a nonempty set U of real numbers that is bounded above is the real number c that satisfies the following properties: 1. c is an upper bound of U. 2. If b is any other upper bound of U, then c ≤ b.

Step-by-step solution

Define the Sets

Define \(U\) as a nonempty set of real numbers that is bounded above. Define \(T\) as the set of upper bounds of \(U\), and \(S\) as the set of real numbers that are not upper bounds of \(U\).

Show that S and T are nonempty sets

Since U is bounded above, this means that there exists some real number that is greater than or equal to every element in U. Therefore, T is nonempty. The set S is also nonempty because U is nonempty (meaning it has at least one element) and that element is not an upper bound of U as it is in the set U.

Apply Dedekind's Theorem

Dedekind's theorem states that: If S and T are nonempty disjoint sets of real numbers such that every element of S is less than every element of T, then there exists a real number \(c\) that is either the largest element of S or the smallest element of T. Since S and T are nonempty, by properties (A)-(H), we have that every element of S is less than every element of T. By Dedekind's theorem, this means that there exists a real number \(c\) that is either the largest element of S or the smallest element of T.

Prove that c is the supremum of U

To show that \(c\) is the supremum of U, we need to prove two things: 1. \(c\) is an upper bound of \(U\). 2. If \(b\) is any other upper bound of \(U\), then \(c \leq b\). To show (1), we consider two cases: Case 1: If \(c\) is the smallest element of T, since every element in T is an upper bound of U, \(c\) is an upper bound of U. Case 2: If \(c\) is the largest element of S, notice that \(c \in S\), meaning \(c\) is not an upper bound of U. Since \(c\) is not an upper bound, there must exist an element \(u \in U\) such that \(u > c\). But since T is the set of upper bounds, there must exist a \(t \in T\) such that \(t \geq u\). Since every real number in S is less than every real number in T, we have \(c \leq t \Rightarrow c \leq u\), which contradicts our assumption that \(u > c\). This contradiction implies that \(c\) must be a minimum of T. Therefore, \(c\) is an upper bound of \(U\). To show (2), let \(b\) be any other upper bound of \(U\). Since \(c\) is the smallest element of T and \(b\) is also in T, we have \(c \leq b\). Since both (1) and (2) are satisfied, we can conclude that \(c\) is the supremum of U.

Key concepts

Real Analysis

Real Analysis is a branch of mathematics that deals with the behavior and properties of real numbers, sequences, series, and functions. In real analysis, the concept of a supremum, or the least upper bound, plays a crucial role in understanding the bounds of sets of real numbers.

When we talk about a set being bounded above, we mean that there is some real number that is greater than or equal to every element of the set. The supremum of a set is the smallest real number that fulfills this criterion. Understanding the supremum helps to establish the completeness property of the real numbers, which states that every nonempty set of real numbers that is bounded above has a supremum.

Dedekind's Theorem

Dedekind's theorem is a pivotal concept in real analysis. It lays down the foundation for the existence of supremums and infimums.

The theorem asserts that if we have two nonempty sets of real numbers, one being S (which consists of numbers not greater than the elements of a particular set), and the other being T (consisting of numbers that are greater than or equal to the elements of the set), and every element of S is less than every element of T, then there will exist a precise number that divides these two sets. This number is either the greatest number in S or the least number in T, establishing a clear boundary between them.

Dedekind's theorem is vital in proving that every set bounded above has a supremum because it guarantees the existence of a definitive splitting point, where no number in S is greater, and no number in T is smaller.

Upper Bounds

In the context of sets, an upper bound is a number that is greater than or equal to every element within a set of numbers. If a set of real numbers is bounded above, it implies the existence of at least one such upper bound.

The upper bounds of any given set can form their own set, known as the set of upper bounds. It's important to note that within the set of upper bounds, there might be a least upper bound, which is called the supremum. This supremum may or may not be a part of the original set but is critical as it marks the lowest level of encapsulation of the set, concerning its maximum limits.

Properties of Real Numbers

The real numbers have several well-defined properties that are essential in the study of real analysis. These include, but are not limited to, order properties, completeness, fields properties, and the existence of limits.

Order properties are what allow us to compare real numbers and say that one is greater, less, or equal to another. These properties are what make it possible to understand concepts such as bounds and extrema (minimum and supremum).

Completeness refers to the property of real numbers that every nonempty set of real numbers that is bounded above has a least upper bound, or supremum. It is this property that the original exercise hinges on and that Dedekind’s theorem helps to elucidate.