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!
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 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 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. 👍
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 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.
![[a,b] is compact](http://i.ytimg.com/vi/DGjYSfu3pr8/mqdefault.jpg)
![[a,b] is compact](/img/tr.png)
This is the true example of quality content
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!
Thank you!!!
speaking of Willem Dafoe: DO YOU KNOW HOW MUCH I SACRIFICED!!!??? + I'm something of a "mathematician" myself.
What a great theorem
So the Heine Borel Theorem proves necessity and sufficiency of closed and boundedness for compactness in euclidian space Rn?
you described total boundedness which is a stronger form of boundedness, but called it boundedness. Am I right?
Thanks for such a great content with love from India
What if initial set is open? Where does the proof fails then?
Subtle. Thank you very much.
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?
There’s a video on that, I think it’s called cantor intersection theorem, it’s non trivial
@@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.]
It’s more complicated than that, check out the video
@@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. 👍
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
It’s not circular, there’s another video proving the nested thing without using compactness
great video! love the willem dafoe bit lol
Can u show me a research video? Thx!
What if none of the sub-boxes have finite subcovers?
One of them will, that’s what we’re proving
@@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.
I wish u explained what compact meant :(
Someone’s not checking out my playlist
Can you tell Dr Peyam what's degree of mathematics which is high level of post PhD?
PhD
Tell me the derivative of cos^x(alpha) wrt x
ln(cosα)*cos(α)^x
Nice👍
It’s a ball in a box LMAO no one got that
The highest math that I've completed is Calc one lmao
Check the subtitles @ 0:06 Lol
Lmao
Willem Dafoe ah ah. That dude is crazy XD
🤣🤣
Subs at 6 seconds🤣🤣
Ryan, how's Oreo and Co going?
Yes not seen oreo since many videos
@@adityadwivedi4412 Thank you. I hope he's okay.
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.
Dude wtf!?That’s plain rude. Also, I’d say he looks rather fine
@@123bluestorm1 I believe Erik, women see and think those sort of things ... and yes, perhaps not appropriate (to publicly state it.)