Real Analysis 10 | Bolzano-Weierstrass Theorem

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This is my video series about Real Analysis. We talk about sequences, series, continuous functions, differentiable functions, and integral. I hope that it will help everyone who wants to learn about it.

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00:00 Intro
00:20 Bolzano-Weierstrass theorem
01:13 Proof
05:42 Credits

#RealAnalysis
#Mathematics
#Calculus
#LearnMath
#Integrals
#Derivatives

I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[javierpicazo]

I'm addicted to the way you pronounce ''real analysis'' at the intro of every video

[punditgi]

Many thanks for this video!
Best explanation of the theorem I have seen anywhere! 👍

[gopinathan]

I don't have any good master to teach me basics of real analysis better than you ... Thank you very much. I'm from India.

[gustavocardenas]

Awesome, this is perhaps the cleanest proof of the BW theorem I've ever seen! Please keep it up!

[qiaohuizhou]

0:25 BW theorem
5:22 every bounded sequence has at least one accumulation point (check the textbook to verify)

[douglasstrother]

True for Complex Numbers, interesting!

[Bulgogi_Haxen]

The only German teacher who explains the mathematics in human understandable language I found.

[herp_derpingson]

This is also binary search :)

I am seeing a pattern. We are showing that if we run binary search long enough, we will converge to some infinitesimally small point, sandwitched between the binary search process.

[luigicamilli]

Thank you so much for this awesome content. I took a Real Analysis class three years ago and I am using your videos to brush up on a bit of proof-based math prior to taking more advanced courses this Fall. Please keep it up. It would be absolutely fantastic if you could add partial differential equations and stochastic calculus crash courses similar in style and delivery to this one.

[oliversc]

Thank you very much for the video, very helpful!!

[cprt.d]

Frage: Du sagst, wir nehmen das linke unendliche Intervall, aber das neue Intervall mit c1 und d1 ist doch die rechte Seite. Das verwirrt mich gerade,

[lbsquat]

I use this theorem to trade and it works beautifully.

[vaginalarthritis]

I enjoyed your inclusion of the sandwich theorem at the end. Only wish you started this Real Analysis series (No pun intended) sooner. I have a final exam tomorrow and your videos are a comfort.

[jonahstrummer]

I wonder is there a proof using the least upper bound property (that every bounded sequence has a least upper bound in the real numbers)?

[willorchard]

Hi sorry for the stupid question, but how do we know that we can choose each a_{n_{k}} such that n_{k+1} > n_{k}? Does this just follow directly from the fact that each new bisection contains infinitely many members - I can see it but I am not sure how to write that intuition down...

[frederickburke]

1:30 how do you know which half contains infinitely many elements of the sequence? That's a rather large step in the proof.

[bhaswatasaikia]

How does thee proof work for a constant sequence?

[i-fanlin]

Could anyone tell me why we can define the subsequence: a_n_k belongs to [c_k, d_k]? See 4:34
Because I think not all bounded sequences have the subsequence, where a_n_k belongs to [c_k, d_k].

[mmanojkumar]

Hey, What book do you follow (and/or suggest) for real analysis part?

[synaestheziac]

Does the definition of a_n_k at the end of the proof rely on the axiom of choice?