Supremum, Infimum: Definition and Explanation

Not all sets of real numbers have a maximum and a minimum. In this video, we introduce the supremum and infimum as a generalization of max and min. Sup and inf appear everywhere in analysis.

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Music: OcularNebula - The Lopez

[BuddyNovinski]

Haydn, You're my inspiration to dive back into the math I never took. You have a nice beard, but you don't need it to hide your youth. Your brilliance outshines your youth. I never had set theory, but it's long overdue I find it out.

[dimitrisanasto]

Very good intuitive introduction to the concept, greetings from Department of Mathematics, Athens Greece,

[MichaelRothwell]

Very nice. When you said every set that has a lower bound has an infimum, I think you meant every non-empty set. Also, you could go on to talk about lim inf and lim sup, but perhaps you'll cover that in another video.

[mohamedibrahim]

Hey we really miss your videos, can you return uploading

[angelmendez-rivera]

Here is another perspective to understanding why we talk about suprema when talking about sets of real numbers. The real numbers form a field, which means that there are operations + and · satisfying the axioms we are familiar with: associativity, commutativity, they have an identity element, distributivity, and invertibility. The rational numbers, though, also form a field. So this raises the question: what distinguishes the real numbers from the rational numbers? How do you get from the rational numbers to the real numbers? In other words, what kind of extension do you use to get from the rational numbers to the real numbers? The answer is given in the video: every bounded set of real numbers has an infimum. This is *not* true with the rational numbers. Consider, for example, the set {q in Q : q·q < 2}. This is a subset of Q, the field of rational numbers, but there is no rational number that is the supremum of this set. On the other hand, if we consider the analogous set, for real numbers, {q in R restricted to Q : q·q < 2}, then there does exist a real number that is the supremum of this subset of R, and that real number is named sqrt(2).

The rational numbers can be built completely, using only 0, 1, addition, additive inverses, multiplication, and multiplicative inverses. You have 0, and you have 1. You can have 1 + 1, and you can have 1 + (1 + 1) = (1 + 1) + 1 = 1 + 1 + 1. You can always keep adding 1 without end, and this always produce a new number. This gives rise to the natural numbers. However, we know that there exists some x such that 1 + x = 0, and we know this because the rational numbers form a field. Such an x is relabeled as –1. So you can have (–1) + (–1), and then (–1) + (–1) + (–1), and so on. This gives you the integers. You can also take the multiplicative inverse of each of the quantities already created. So you can have (1 + 1)^(–1), (1 + 1 + 1)^(–1), [(–1) + (–1)]^(–1), and so on. You can multiply these new quantities by the old ones. Each of these multiplications is a rational number. And this actually gives you every rational number.

The real numbers can be constructed in the exact same way, but to get the real numbers, there is one more step. Now that you have all of these products of the form (m + m + ••• + m)·(n + n + ••• + n), where m, n = –1, 1, you can take the set of all such products, and find the subsets that are bounded from below. We will know introduce an axiom: we will axiomatically declare that each of those subsets has an infimum. We may not know what that infimum is or how to express it using typical notation, but what we are saying is that this infimum must exist, because our axioms require it. So each such subset has an infimum, and that infimum is a real number. Every real number that had not already been constructed has now been postulated to exist as the infimum of some set. Every real number that has no equivalent in the rational numbers (a.k.a the irrational numbers) is the infimum of some set of rational numbers. And this is how you get the rational numbers extended into the real numbers: by declaring that every bounded subset *must* have an infimum.

This is an important detail, because many people think of the real numbers as just being a bunch of nonterminating strings of digits represented in base 10. That is not how it actually is. While we do use these strings to represent approximations of those real numbers, we are not required to do so. The strings could be in base 2, base 3, or base e, for all we care about. In fact, the notation we use to represent the numbers is completely irrelevant, and does not itself change how the numbers are defined. π is π, regardless of whether we represent its approximations using base 10, base 2, or whether we use an integral representation of it instead, or whether we use series to represent it instead. How a quantity is defined, as the supremum or infimum of some set of rational numbers, does not become affected by how it is represented to a human. The quantity still exists as a real number, and is still ultimately well-defined. Whether we can represent the quantity in a nice, useful notation, or not, is an entirely different question altogether. This is why it is wrong to think of real numbers as "the decimal numbers." It treats a notation system as being itself a mathematical structure, when that is just not accurate. The mathematical structure does not care about how you choose to represent it. As long as the axioms are consistent, the structure works the way it does, and that is that.

[Jack-cmch]

Why even bother with saying "greatest lower bound".... just say lowest possible number in the set. Math is made artificially complicated