The Bolzano Weierstraß Theorem

Bolzano-Weierstrass Theorem

Welcome to one of the cornerstone theorems in Analysis: The Bolzano-Weierstraß Theorem. It is the culmination of all our hard work on monotone sequences, and we'll use this over and over again in this course. Luckily the proof isn't very difficult, since we've done all the hard work in previous videos.

[FT]

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)!

[Domzies]

I've literally just took a break from revising sequences after running into this theorem.

[Robinzon__Kruzo]

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.

[jordia.]

I very much appreciate your enthusiasm 💪

[diskasiTv]

More videos please really inspiring for a non math major

[michellauzon]

If a <= Sn <= b, then divide (a, b) in two equal parts. One part has an infinite number of Sn. Divide this part in two and so on ...

[tehdii]

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.

[bilubilu]

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

[HighLordSythen]

Cool. I'm pretty sure this is something I've never seen before.

[henrebooysen]

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?

[Fematika]

This is the same as saying that the closed interval is subsequence compact! Or, equivalently, that it is compact.

[kalles]

Bolzano-Weierstraß-Theorem however says nothing about the value of the limit of the subsequence. It only says, that the limit exists.

[leswhynin]

Here's the first sequence of words for this video

[meiwinspoi]

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.

[alexcsouza]

Nossa que susto! achei que o título era "Teorema do Bolsonaro e Weintraub".