Real Analysis | The Supremum and Completeness of ℝ

We look at the notions of upper and lower bounds as well as least upper bounds and greatest lower bounds of sets of real numbers. We also prove an important classification lemma of least upper bounds. Finally, the completeness axiom of the real numbers is presented.

[gustafa]

You're a legend, dude. I'm relearning this subject right now on my own.

[GnarGnaw]

Supremum, or sometimes we call it *soup*

[Blure]

I'm learning real analysis on my own, your videos are incredibly helpful!

[JohnWick-xdzu]

This channel is pure gold!!! Thank you prof. Penn <3 sending love from Egypt

[andrewcoakley]

Wow thank you! Currently taking a real analysis class and this was the clearest and most concise explanation that I've heard on this topic. Cheers.

[tomatrix]

I am learning Real Analysis on my own and this is really good. Thanks so much because otherwise I’d be sort of astray with no clear path

[thobilesikakane]

Very good detailed and to the point. Thank you so much. Saved me hours of non-productive studying.

[dozenazer]

You’re just in time. I’ve just learned that thing on calculus. Thank you so much for the proof.

[iridium]

this is amazing, this is by far the most underrated math channel on youtube, i am still pretty young but i am passionate about mathematics, and this channel has helped me alot, thank you so much for this.

[maxpercer]

A useful theorem:
For a non empty set S, given any epsilon greater than zero,
if S is bounded above there exists an A in S such that sup(S) - e < A ≤ sup(S),
and if S is bounded below there exists a B in S such that inf(S) ≤ B < inf(S) + e.

[vbcool]

Awesome!


Please also cover Axiom of Choice and ZFC Set Theory if possible.

[robson]

I've just found your channel, and man, it is a blessing! Keep up this amazing work!

[scipionedelferro]

Wow!! You're great, best math channel on YouTube! The very best was the last example, with the sequence of digits of pi. It shows that even the algebraic numbers are not complete, as pi is transcendental. And the completeness is an axiom of the Real numbers. But there are countable sets that are complete too. So it is not this axiom that requires the Reals to be uncountable. Must be other axioms...

[silversky]

so blessed to find you at the start of my real analysis course

[criptonessy]

Wow great video, I think I'll need to watch it a second time to fully digest it, but very clear, thank you!

[schalkzijlstra]

Thanks man, all the way from South Africa

[jabronimargaretti]

god bless you man. thanks for doing these lectures!

[PunmasterSTP]

Completeness? More like "You need this"...if you're taking a real analysis course! Thanks again for making and sharing all these wonderful videos.

[Anna-jycj]

This is great, I've been looking for something like this and bam here you are .

[Behroozifyable]

Fantastic, but at the very end you forgot to put a line through 'belongs to' sign as you say pie doesn't belong to rational numbers.