the pen and scissors analogy is absolutely brilliant! In my analysis class we defined this theorem as "every bounded set that's a subset of \mathbb{R}^N with infinitely many elements must have at least 1 accumulation point". After some thought I think I see how it's equivalent-- in every single dimension, we can use the proof in the video to get a convergent subsequence, and then mash them all together to get the accumulation point (which always has a neighbor in each dimension due to the converging subsequence)!
I like this prove rather than a prove by bisection method, because prove by bisection method is harder to formalize than this prove. Thank you for demonstrating this method. I didn't heard about monotonic subsequence existence theorem at my university.
Thanks for the video, I'm studying calculus with theory and starting real analysis, this is an foundational result on the field for certain. Success for you man
I am reading David Foster Wallace Histrory of infinity and thanks to youtube I can now visualize every thing he is writing about ;) Your video is one of the simplest and the best to understant intuitively this concept. Thank you.
I really like this proof. A lot simpler and easier to understand than some other proofs I have seen. But which theorem is used to show that any bounded sequence must have a monotonic subsequence?
you being bounded by a tie now a days. diverge to shorts and T. Jokes apart - what if the lower and upper bounds themselves are ever increasing - like in x.sin x. will this BW theorm be valid.
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the pen and scissors analogy is absolutely brilliant!
In my analysis class we defined this theorem as "every bounded set that's a subset of \mathbb{R}^N with infinitely many elements must have at least 1 accumulation point". After some thought I think I see how it's equivalent-- in every single dimension, we can use the proof in the video to get a convergent subsequence, and then mash them all together to get the accumulation point (which always has a neighbor in each dimension due to the converging subsequence)!
Long awaited, glad you got us there!
I like this prove rather than a prove by bisection method, because prove by bisection method is harder to formalize than this prove. Thank you for demonstrating this method. I didn't heard about monotonic subsequence existence theorem at my university.
I very much appreciate your enthusiasm 💪
Thanks for the video, I'm studying calculus with theory and starting real analysis, this is an foundational result on the field for certain. Success for you man
I've literally just took a break from revising sequences after running into this theorem.
I am reading David Foster Wallace Histrory of infinity and thanks to youtube I can now visualize every thing he is writing about ;) Your video is one of the simplest and the best to understant intuitively this concept. Thank you.
More videos please really inspiring for a non math major
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I really like this proof. A lot simpler and easier to understand than some other proofs I have seen. But which theorem is used to show that any bounded sequence must have a monotonic subsequence?
It’s in the description I believe
This is the same as saying that the closed interval is subsequence compact! Or, equivalently, that it is compact.
Cool. I'm pretty sure this is something I've never seen before.
That expert Aussprache!!!
Bolzano-Weierstraß-Theorem however says nothing about the value of the limit of the subsequence. It only says, that the limit exists.
Here's the first sequence of words for this video
Nice and Cauchy
Can you prove it in R^n
See description
Thank you very much. Khalee mamnoon.
you being bounded by a tie now a days. diverge to shorts and T. Jokes apart - what if the lower and upper bounds themselves are ever increasing - like in x.sin x. will this BW theorm be valid.
Mei wins Poi I wouldn’t have thought of that but now that you’ve mentioned it it makes sense to me.
That would be an example of an unbounded sequence though. There would be a different sequence dealing with that.
Nossa que susto! achei que o título era "Teorema do Bolsonaro e Weintraub".