Order theory | Articles containing proofs | Real analysis
In mathematics, the least-upper-bound property (sometimes called completeness or supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X. Not every (partially) ordered set has the least upper bound property. For example, the set of all rational numbers with its natural order does not have the least upper bound property. The least-upper-bound property is one form of the completeness axiom for the real numbers, and is sometimes referred to as Dedekind completeness. It can be used to prove many of the fundamental results of real analysis, such as the intermediate value theorem, the Bolzano–Weierstrass theorem, the extreme value theorem, and the Heine–Borel theorem. It is usually taken as an axiom in synthetic constructions of the real numbers, and it is also intimately related to the construction of the real numbers using Dedekind cuts. In order theory, this property can be generalized to a notion of completeness for any partially ordered set. A linearly ordered set that is dense and has the least upper bound property is called a linear continuum. (Wikipedia).
Math 101 091517 Introduction to Analysis 07 Consequences of Completeness
Least upper bound axiom implies a "greatest lower bound 'axiom'": that any set bounded below has a greatest lower bound. Archimedean Property of R.
From playlist Course 6: Introduction to Analysis (Fall 2017)
Least Upper Bound Property In this video, I state the least upper bound property and explain what makes the real numbers so much better than the rational numbers. It's called Real Analysis after all! Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZ
From playlist Real Numbers
Math 101 091317 Introduction to Analysis 06 Introduction to the Least Upper Bound Axiom
Definition of the maximum (minimum) of a set. Existence of maximum and minimum for finite sets. Definitions: upper bound of a set; bounded above; lower bound; bounded below; bounded. Supremum (least upper bound); infimum (greatest lower bound). Statement of Least Upper Bound Axiom (com
From playlist Course 6: Introduction to Analysis (Fall 2017)
Proof of the Least Upper Bound Property In this video, I present a very elegant proof of the least upper bound property. This proof really illustrates why Dedekind cuts are so nice. Least Upper Bound Property: https://youtu.be/OQ0HBjq8OWE Dedekind Cuts: https://youtu.be/ZWRnZhYv0G0 Che
From playlist Real Numbers
Math 131 090516 Lecture #02 LUB property, Ordered Fields
Least Upper Bound Property and Greatest Lower Bound Property; Fields; Properties of Fields; Ordered Fields and properties; description of the real numbers (ordered field with LUB property containing rational numbers as subfield); Archimedean property #fields #orderedfields #leastupperboun
From playlist Course 7: (Rudin's) Principles of Mathematical Analysis
Upper and Lower Bound In this video, I define what it means for a set to be bounded above and bounded below. This will be useful in our definition of inf and sup. Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmCZggpJZvUXnUzaw7fHCtoh
From playlist Real Numbers
Monotone Sequence implies Least Upper Bound
Monotone Sequence Theorem implies Least Upper Bound Property In this video, I prove a very interesting analysis result, namely that the Monotone Sequence Theorem is EQUIVALENT to the Least Upper Bound Property. This makes the least upper bound property more intuitive, in my opinion. Chec
From playlist Sequences
Math 101 Introduction to Analysis 091815: Least Upper Bound Axiom
The least upper bound axiom. Maximum and minimum of a set of real numbers. Upper bound; lower bound; bounded set. Least upper bound; greatest lower bound.
From playlist Course 6: Introduction to Analysis
Axiomatics and the least upper bound property (I1) | Real numbers and limits Math Foundations 121
Here we continue explaining why the current use of `axiomatics' to try to formulate a theory of `real numbers' is fundamentally flawed. We also clarify the layered structure of the rational numbers: we have seen these several times already in prior discussion of the Stern- Brocot tree, her
From playlist Math Foundations
Real Analysis Chapter 1: The Axiom of Completeness
Welcome to the next part of my series on Real Analysis! Today we're covering the Axiom of Completeness, which is what opens the door for us to explore the wonderful world of the real number line, as it distinguishes the set of real numbers from that of the rational numbers. It allows us
From playlist Real Analysis
Lecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property
MIT 18.100A Real Analysis, Fall 2020 Instructor: Dr. Casey Rodriguez View the complete course: http://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP61O7HkcF7UImpM0cR_L2gSw Finishing the lecture on Cantor’s notion of
From playlist MIT 18.100A Real Analysis, Fall 2020
Fundamentals of Mathematics - Lecture 20: Infinite Intersections and Least Upper Bound Property
course page: https://www.uvm.edu/~tdupuy/logic/Math52-Fall2017.html worksheets - DZB, Emory videography - Eric Melton, UVM
From playlist Fundamentals of Mathematics
Lecture 4: The Characterization of the Real Numbers
MIT 18.100A Real Analysis, Fall 2020 Instructor: Dr. Casey Rodriguez View the complete course: http://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP61O7HkcF7UImpM0cR_L2gSw An introduction to properties of fields and
From playlist MIT 18.100A Real Analysis, Fall 2020
Calculating With Upper & Lower Bounds | Number | Maths | FuseSchool
Calculating With Upper & Lower Bounds | Number | Maths | FuseSchool In this video we are going to look at how to calculate with upper and lower bounds. To find the upper bound of an addition or of an area, you would want to multiply the upper bounds of both measurements, as this would g
From playlist MATHS: Numbers