Introductory Real Analysis, Lec 3: Irrational Numbers, Supremums, Completeness, Sqrt(2) Exists

Bill Kinney, Bethel University Department of Mathematics and Computer Science. St. Paul, MN.

(0:00) 0.9999... = 1 is true.
(6:25) sqrt(3) is irrational.
(8:01) sqrt(6) is irrational.
(10:33) sqrt(2) + sqrt(3) is irrational.
(12:55) Statement of the completeness axiom.
(14:31) Definition of what it means for a number to be an upper bound of a set of real numbers.
(17:48) Definition for a set to be bounded above.
(19:46) Examples.
(20:42) Definition of the supremum of a set of real numbers.
(22:55) Examples of sups.
(27:54) Visual description of the proof that every nonempty set of real numbers that is bounded below has an inf.
(30:19) Scratch work details of how to think about the proof that sqrt(2) exists using the completeness axiom.
(49:09) Definition of what it means for a set to be dense in the set of real numbers R.
(51:32) The set of rational numbers Q is dense in the set of real numbers R.

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[RedShiftedDollar]

Finally a proof of the irrationality of sqrt2 that acknowledges that the classic "assume rational then disprove by contradiction" only allows you to prove that it is not rational(simple negation of the assumption). An additional logical step involving proof of existence is required to then distinguish between: is not rational and is not irrational(i.e. Does not exist), and is irrational (exists and is not rational therefore is irrational). Your lecture is one of the best on the internet on this subject. Thanks for sharing.

[sofiavelosomagioliemello]

My gratitude for your work is infinite. Thank you, thank you, thank you.

[tekaaable]

Finally summer vacation. Now I can finally spend my time doing what I want to do. That is start up on real analysis :)

[zeropoints]

You're a great lecturer! I'm taking a calculus course but the classes are super tedious. Your style is infinitely more engaging.

[pablensemarin]

This classes have been of great help during my first college online semester, thank you!

[LuqfCS]

these videos are amazing - your explanations are clear, and easy to understand. thanks for making these resources available to the public!

[darstking]

Hi Professor. Not sure if you are still monitoring this, but I just want to say thank you. These lectures really are fantastic.

One question -- is there any way you could post/show the homework assignments? It would be great if I could not only follow along but also work through some exercises.

Thanks Professor

[christineh]

This is so incredibly helpful! Thank you so much!

[michaellewis]

Ah that’s a nice one basically if we assert have something greater we know there’s a number smaller, if we assert something smaller, we get something greater, so we’re left with what’s in btwn that is, beta=sqrt(2) exists by elimination w/the trichotomy principe.

[wernerhartl]

X=.999, 10x=9.990, 10x-x=8.991=9-.009, x=1-.001. In general, x=1-10^n. You can’t subtract .999… from 10.999… because subtraction starts from the right. But Lim .999… =1. Or LUB .999…=1

[Mycrosss]

How is x-2 positive for sure, but you had to prove that x+2 was positive by adding the rule of x>=0? Wouldn't x-2 < x+2 be obvious out of all the base rules which would lead into positive < x => x is positive? And how did you decide that x-2 is positive, I can't seem to understand :( . Thanks for the lectures, they really help a lot though !

[jonahellison]

I have been practicing how to 'think like a mathematician', this involves using the tools I have gained from mathematics, and creating example questions which I can solve. I am rather new to this type of thinking, can you critique my question, on what I may need to incorporate into my thinking process to acquire reasonable questions with supporting proofs?

For one example question, I have asked myself:
For all x contained in R^n, and x contained in R^1 and R^2 and so forth, in a continuous manner. Can I differentiate R^1 from R^2, such that x is greater than or less than, as a variable to the previous or preceding geometric space.

Essentially, I may tackle such a question by using ordered fields and the supremum + infimum.

[BillShillito]

Has your son ever heard of hyperreal numbers? :)

[AubreyForever]

I understand the textbook is horrible from a few reviews.