Heine Borel Theorem

Here I prove the Heine-Borel Theorem, one of the most fundamental theorems in analysis. It says that in R^n, all boxes must be compact. The proof itself is very neat, and uses a bisection-type argument. Enjoy!
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[@umerfarooq4831]

This is the true example of quality content

[@annakbanana861]

Thank you so much for this video! We went over the Heine-Borel Theorem in my Analysis class a couple days ago, but I still had trouble understanding some concepts. I kept pausing, taking notes, and repeating what you said to make sense of it; it does now! I really appreciate it!

[@drpeyam]

Thank you!!!

[@theproofessayist8441]

speaking of Willem Dafoe: DO YOU KNOW HOW MUCH I SACRIFICED!!!??? + I'm something of a "mathematician" myself.

[@Nick-kg7sk]

What a great theorem

[@dgrandlapinblanc]

Subtle. Thank you very much.

[@shivaudaiyar2556]

Thanks for such a great content with love from India

[@usrrrrrrr5677]

great video! love the willem dafoe bit lol

[@kerljenge4625]

Great video! But I think the argument that boxes are compact using FIP is circular. The arbitrary intersection of the nested sequence is only nonempty if the box is compact to begin with

[@drpeyam]

It’s not circular, there’s another video proving the nested thing without using compactness

[@Kaassap]

So the Heine Borel Theorem proves necessity and sufficiency of closed and boundedness for compactness in euclidian space Rn?

[@Kaassap]

you described total boundedness which is a stronger form of boundedness, but called it boundedness. Am I right?

[@vadimpavlov6037]

What if initial set is open? Where does the proof fails then?

[@nnaammuuss]

19:22 Finite intersection property?? I thought that was just the De Morgan dual of the open-cover/finite-subcover thing!! If you're gonna assume that.. 🙄
😊 You may wanna extract a cauchy sequence or appeal to completeness or something.. or, is that the part you're refering to as finite intersection property?

[@drpeyam]

There’s a video on that, I think it’s called cantor intersection theorem, it’s non trivial

[@nnaammuuss]

@@drpeyam Is it? Once you have a decreasing sequence of closed sets with shrinking diameter (shrinking to zero, I mean), I thought it was pretty obvious that if you pick a point from each set, it's going to form a cauchy sequence! And once you have a limit, it will surely be a limit point of each set, and so in the whole intersection. Am I missing something?
[Edit: typo.]

[@drpeyam]

It’s more complicated than that, check out the video

[@nnaammuuss]

@@drpeyam managed to find it, thank you 😊. It seems to replace ‘diameter shrinking to zero’ with just ‘bounded’, which forces you to pick a subsequence (going to, say, the lim sup) in one dimension, and subsequence of subsequence of subsequence etc, in multi-dimension. However, when the diameter shrinks to zero all that ain't required, and what I wrote above essentially appears to be correct (except that the limit may be a _point_ or a limit point of each F_n, in case the chosen sequence was eventually constant or something).
Anyway, 😊 👍 congrats, you appear to be putting up a very good series on mathematics-much closer to the ‘‘real stuff’ done in higher academia. I myself, since the beginning of the pandemic thought of doing something like this (but I was too lazy to lift my ass off the chair)-I would call it the GAGA Series-that includes series of undergraduate/graduate/PhD courses eventually guiding itself towards the GAGA theorems, just as an aim, but really a tour of mathematics otherwise. I knew Borcherds was also up to something like that, but as he was old, and never gets up from his chair, I thought I'd have some physical advantage to sing and dance. But as much younger people are taking up the same torch, I'm wondering if I should do it at all. But, 😆 my best, very best wishes man, go on, and go far. 👍

[@SuperMaDBrothers]

It’s a ball in a box LMAO no one got that

[@lacasadeacero]

Can u show me a research video? Thx!

[@SuperMaDBrothers]

I wish u explained what compact meant :(

[@drpeyam]

Someone’s not checking out my playlist

[@hybmnzz2658]

💥🔥🔥🔥🔥💪⏯▶️

[@erwinreha6723]

Nice👍

[@geraltofrivia9424]

Willem Dafoe ah ah. That dude is crazy XD

[@drpeyam]

🤣🤣

[@toaj868]

What if none of the sub-boxes have finite subcovers?

[@drpeyam]

One of them will, that’s what we’re proving

[@toaj868]

@@drpeyam Sorry I realised my mistake. Even if the other sub-boxes did not have finite subcovers (before proving that they do), we could just do the same thing with the other sub-boxes as well.

[@abs0lute-zer061]

The highest math that I've completed is Calc one lmao

[@koenth2359]

Check the subtitles @ 0:06 Lol

[@pierfrancescopeperoni]

Lmao

[@rikhalder5708]

Can you tell Dr Peyam what's degree of mathematics which is high level of post PhD?

[@drpeyam]

PhD

[@marco-vz5kv]

Tell me the derivative of cos^x(alpha) wrt x

[@connorshea9085]

ln(cosα)*cos(α)^x

[@geetathakur445]

Subs at 6 seconds🤣🤣

[@gheffz]

Ryan, how's Oreo and Co going?

[@adityadwivedi4412]

Yes not seen oreo since many videos

[@gheffz]

@@adityadwivedi4412 Thank you. I hope he's okay.

[@Iarlen]

I was trying to show your content to my girlfriend and she said you should "shave your unibrow"...
Then she told me to tell you this in the comments.

[@123bluestorm1]

Dude wtf!?That’s plain rude. Also, I’d say he looks rather fine

[@gheffz]

@@123bluestorm1 I believe Erik, women see and think those sort of things ... and yes, perhaps not appropriate (to publicly state it.)