Direct Bolzano Weierstraß

Bolzano-Weierstrass Theorem (Direct Proof)

In this video, I present a more direct proof of the Bolzano-Weierstrass Theorem, that does not use any facts about monotone subsequences, and instead uses the definition of a supremum. This proof is taken from Real Mathematical Analysis by Pugh, and its advantage is that it just uses first principles. Enjoy!

[darkseid]

Just woke up . Good way to start a morning with math .

[elephantdinosaur]

That was one of the most direct proofs of the Bolzano Weierstrass theorem I've seen and only 12 mins long!! The only thing you really assumed was the the LUB property of the reals!!! Nice going Peyam! 😎

I think liminf x_n = c if my intuition is right 🤔

[dgrandlapinblanc]

Ok. I understand now. It's very subtle. Thanks.

[pierreabbat]

He lived in Prague, but his native language was German and his surname is Italian, so <z> is pronounced /ts/.

[MrImwolff]

brining me back to my real analysis course

[ashwinvishwakarma]

but you need to show c+epsilon is in your set to say c+epsilon < c. Also I think you meant to say c - epsilon is an upperbound, not c+epsilon, then you use the sup property

[DanielCastro-uylu]

I know is something totally different but, talking about Weierstrass, what about a video talking about Stone-Weierstrass theorem? 👀

[dgrandlapinblanc]

Ok. Thank you very much but i don't understand this point where they are two negations at the final of the video. A negation of a negation is an affirmation isn't it ? So when you prove the contradiction you prove that doesn't work. Sorry for my answer.

[louispoulain]

Take x_n=(-1)^n sequence in [a, b]=[-1, 1]. Isn't your C empty in this case ? This is a problem isn't it ?