In mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is a greatest element in that is less than or equal to each element of if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to each element of if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB). The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are close to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the set of positive real numbers (not including ) does not have a minimum, because any given element of could simply be divided in half resulting in a smaller number that is still in There is, however, exactly one infimum of the positive real numbers relative to the real numbers: which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. An infimum of a set is always and only defined relative to a superset of the set in question. For example, there is no infimum of the positive real numbers inside the positive real numbers (as their own superset), nor any infimum of the positive real numbers inside the complex numbers with positive real part. (Wikipedia).
Definition of Supremum and Infimum of a Set | Real Analysis
What are suprema and infima of a set? This is an important concept in real analysis, we'll be defining both terms today with supremum examples and infimum examples to help make it clear! In short, a supremum of a set is a least upper bound. An infimum is a greatest lower bound. It is easil
From playlist Real Analysis
Proof: Supremum and Infimum are Unique | Real Analysis
If a subset of the real numbers has a supremum or infimum, then they are unique! Uniqueness is a tremendously important property, so although it is almost complete trivial as far as difficulty goes in this case, we would be ill-advised to not prove these properties! In this lesson we'll be
From playlist Real Analysis
Epsilon Definition of Supremum and Infimum | Real Analysis
We prove an equivalent epsilon definition for the supremum and infimum of a set. Recall the supremum of a set, if it exists, is the least upper bound. So, if we subtract any amount from the supremum, we can no longer have an upper bound. The infimum of a set, if it exists, if the greatest
From playlist Real Analysis
Supremum of a set In this video, which is the most important video of the chapter, I define the supremum of a set of real numbers. It is like a maximum, except that it always exists, and will be super useful in the rest of our analysis adventure. Check out my Real Numbers Playlist: https
From playlist Real Numbers
Infimum of a set In this video, I define sup's little cousin, the infimum of a set. It is like a minimum, except that it always exists. Supremum of a set: https://youtu.be/lZEcsOn6qUA Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZggpJZvUXnUzaw7f
From playlist Real Numbers
Proof: Supremum of {1/n} = 1 | Real Analysis
The supremum of the set containing all reciprocals of natural numbers is 1. That is, 1 is the least upper bound of {1/n | n is natural}. We prove this supremum in today's real analysis lesson using the epsilon definition of supremum! Definition of Supremum and Infimum of a Set: https://ww
From playlist Real Analysis
Proof: Infimum of {1/n} = 0 | Real Analysis
The infimum of the set containing all reciprocals of natural numbers has an infimum of 0. That is, 0 is the greatest lower bound of {1/n | n is natural}. We prove this infimum in today's real analysis lesson using the Archimedean Principle, which tells us that given any real number x, we c
From playlist Real Analysis
Proof: Supremum of {n/(n+1)} = 1 | Real Analysis
Today we prove the supremum of {n/(n+1)} is 1, using the Archimedean principle and the epsilon definition of supremum of a set. The Real Analysis Playlist: https://www.youtube.com/playlist?list=PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli Proof of Archimedean Principle: https://youtu.be/sG1FHUPdm
From playlist Real Analysis
Supremum of the Union of Sets | Real Analysis
If A and B are two bounded and nonempty subsets of the real numbers, then what is the supremum of their union? What is supAUB? Well, if A and B are bounded nonempty subsets of the reals then we know they both have supremums by the completeness axiom. Then, as it turns out, the supremum of
From playlist Real Analysis Exercises
What is a Riesz Space? -- MathMajor Seminar
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From playlist MathMajor Seminar
inf(S) = -sup(-S) In this video, I present a neat identity relating inf and sup. This means that, from now on, everything that we say for sup will hold for inf as well. Moreover, using this, we can prove the greatest lower bound property. Enjoy! Check out my Real Numbers Playlist: https:
From playlist Real Numbers
sup(A+B) = sup(A) + sup(B) In this video, I use the definition of sup to show that sup(A+B) = sup(A) + sup(B). It's a sup-er awesome identity! Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZggpJZvUXnUzaw7fHCtoh
From playlist Real Numbers
MAST30026 Lecture 13: Metrics on function spaces (Part 1)
I defined the sup metric on the function space Cts(X,Y) where X is compact and Y is a metric space, and proved that the associated metric topology agrees with the compact-open topology. Lecture notes: http://therisingsea.org/notes/mast30026/lecture13.pdf The class webpage: http://therisi
From playlist MAST30026 Metric and Hilbert spaces
Real Analysis - Part 6 - Supremum and Infimum [dark version]
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From playlist Real Analysis [dark version]
Detailed Proof of the Monotone Convergence Theorem | Real Analysis
We prove a detailed version of the monotone convergence theorem. We'll prove that a monotone sequence converges if and only if it is bounded. In particular, if it is increasing and unbounded, then it diverges to positive infinity, if it is increasing and bounded, then it converges to the s
From playlist Real Analysis
Prove Infimums Exist with the Completeness Axiom | Real Analysis
The completeness axiom asserts that if A is a nonempty subset of the reals that is bounded above, then A has a least upper bound - called the supremum. This does not say anything about if greatest lower bounds - infimums exist for sets that are bounded below, but we can use the completenes
From playlist Real Analysis Exercises