Least-upper-bound property

In mathematics, the least-upper-bound property is an important idea that helps us understand the real numbers. This property says that for certain collections of numbers, there is always a smallest number that is bigger than every number in the collection. This special number is called the least upper bound or supremum.

Not all number systems have this property. For example, the set of rational numbers does not always have a least upper bound. This is one reason why mathematicians work with real numbers, which do have this property.

The least-upper-bound property is closely related to the completeness axiom for real numbers. It helps prove many important results in real analysis, such as the intermediate value theorem and the Bolzano–Weierstrass theorem. It also plays a key role in building the real numbers using special methods called Dedekind cuts.

Statement of the property

Statement for real numbers

Let's think about a group of real numbers, like temperatures or scores. If we have a set of numbers, we can find a number that is bigger than every number in that set—this is called an upper bound. For example, if our numbers are 1, 2, and 3, then 4 is an upper bound because it's bigger than all of them.

Now, the least upper bound (or supremum) is the smallest number that is still bigger than every number in the set. Using our example, 4 is an upper bound, but so is 3.5, 3.1, and even 3. The smallest of these is 3, which is the least upper bound.

The least-upper-bound property tells us that if a group of real numbers has an upper bound, then it must also have a least upper bound. This helps us understand how numbers fit together.

Generalization to ordered sets

Main articles: Completeness (order theory) and Bounded complete poset

We can also talk about upper bounds and least upper bounds with other collections that have a way to compare them, not just numbers. For example, think of a group of words where we can say which word comes before another in alphabetical order. If every group of words that has a "top" word (an upper bound) also has a smallest "top" word (a least upper bound), then this collection has the least-upper-bound property.

However, this doesn’t always work. Take the set of rational numbers (numbers that can be written as fractions). Some groups of these numbers have an upper bound but no least upper bound. For example, the numbers that squared are less than or equal to 2 have an upper bound, but their least upper bound isn’t a rational number—it’s the square root of 2, which is irrational. This idea helped mathematicians create the set of all real numbers by filling in these gaps.

Proof

The least-upper-bound property is closely related to other ideas about how real numbers work, like the completeness axiom and the behavior of Cauchy sequences. Depending on how we build the real numbers, this property can either be accepted as a basic rule or shown to be true using other facts.

One way to show this property is by using Cauchy sequences. If we assume that every Cauchy sequence of real numbers comes together to a single point, we can use this to find the smallest number that is still bigger than every number in a set. We start with a simple set and then build two sequences that get closer and closer to each other. In the end, these sequences meet at a point that is the least upper bound of the original set.

Applications

The least-upper-bound property of the real numbers R helps prove important theorems in real analysis.

One key theorem is the intermediate value theorem. It says that if you have a continuous function on a closed interval and the function takes different signs at the ends, then it must cross zero somewhere in between. This can be shown using the least-upper-bound property.

Another important theorem is the extreme value theorem. It states that a continuous function on a closed interval always reaches its highest and lowest values. This is also proved using the least-upper-bound property.

The Heine–Borel theorem tells us that any collection of open sets that covers a closed interval must contain a smaller finite collection that still covers the entire interval. This theorem also relies on the least-upper-bound property.

History

The idea of the least-upper-bound property was first noticed by Bernard Bolzano in 1817. In his paper titled Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege, Bolzano showed how important this property is in mathematics. This property helps us understand how real numbers work and fit together.