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

[@MichaelRothwell1]

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.

[@BuddyNovinski]

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[@dimitrisanasto1168]

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

[@mohamedibrahim1023]

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[@user-lr8od4uz1n]

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[@leeshaocheng239]

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[@angelmendez-rivera351]

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.

[@MichaelRothwell1]

There are several ways to complete the rationals to construct the reals. The most elegant method is to use the Dedekind cut method, as you suggest, because of its simplicity and base-independence. However, completing the reals using infinite decimal expansions (as long as you take appropriate equivalence classes, or simply exclude those ending in an infinite sequence of 9s) is an equally valid, if less elegant, way to do so. I have a book on real analysis that successfully uses the latter approach.

[@angelmendez-rivera351]

@@MichaelRothwell1 I am aware that you can use Cauchy completeness as a metric space to define the real numbers instead. This only proves my point, though: base 10 is just a representation. It makes no difference whether you use base 2 or base 10 or base Graham's number. In fact, you are not even required to use a positional, digital system of notation for representation: you could use whatever complicated scheme you like, as long as it involves sequences of rational numbers. It makes no difference, because at the end of the day, you are always going to repartition the set of sequences under a different equivalence relation, the Cauchy equivalence, to ensure that you have Cauchy sequences, and this is precisely what makes thinking of real numbers as "decimals" not the conceptually appropriate way of thinking about them. It would be appropriate if the representation did actually matter.

[@angelmendez-rivera351]

@@MichaelRothwell1 Also, on that note, since the metric on the rational numbers is not unique, the Cauchy completion is depends on the metric, and is not unique either. As such, there is nothing that stops me from extending the rational numbers in a different direction, such as the p-adic numbers. This is a perfectly valid extension, but it is not the one we are looking for, and so, it produces this awkward scenario where there is no sense in which it is particularly natural and non-arbitrary to go from the rational numbers to the real numbers. The order-theoretic definition solves this problem. That is not to say that this is not a valid construction. I agree with you. But here is my point: my point is that it being valid does not mean it is the conceptually appropriate way to think about it. I mean, this is a lot like that discussion of whether we should adopt category theory as the basis for mathematical foundations. Set theory is a valid way to formalize mathematics, but it is not necessarily the one we should use.

[@adityadwivedi4412]

How are you brother?

[@Jack-cm5ch]

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