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)
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)
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
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
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
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
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
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
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
Real Analysis Ep 3: The Axiom of Completeness
Episode 3 of my videos for my undergraduate Real Analysis course at Fairfield University. This is a recording of a live class. This episode is about the completeness axiom for the real numbers. Class webpage: http://cstaecker.fairfield.edu/~cstaecker/courses/2020f3371/ Chris Staecker we
From playlist Math 3371 (Real analysis) Fall 2020
Real Analysis | The density of Q and other consequences of the Axiom of Completeness.
We present three results that follow from the completeness of the real numbers. 1. The Nested Interval Theorem 2. The Archimedean Principal 3. The density of the rational numbers in the real numbers. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal W
From playlist Real Analysis
The Intermediate Value Theorem, Continuity, and The Axiom of Completeness | Nathan Dalaklis
The Intermediate Value Theorem is something that I said I covered, but apparently I hadn't yet. So here I tackle a proof of the IVT with the help of Continuity and the Axiom of Completeness so that when it does come up again, I can reference it and actually have something to point at :D _
From playlist The New CHALKboard
Nested Interval Property and Proof | Real Analysis
We introduce and prove the nested interval property, or nested interval theorem, or NIP, whatever you like to call it. This theorem says that, given a sequence of nested and closed intervals, that is, closed intervals J1, J2, J3, and so on such that each Jn contains Jn+1, this infinite seq
From playlist Real Analysis
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
WHY DOES THE SQUARE ROOT EXIST IN THE REAL NUMBERS: How to prove that the mth root exists | Nathan D
This week we focus on how to prove that the mth root exists (I realize that most folks would say "nth root" here but I decided to change it up a little bit.) In doing so we will inadvertently give the answer to the question of "why does the square root exist in the real numbers?" for a pos
From playlist The New CHALKboard