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 🤔
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
@@drpeyam I meant because c+epsilon can be greater than b. The set is only concerned with x in [a,b]. The way I understood it was since x_n for n>N, is not in (c-e, c+e), and we know c-e
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.
Just woke up . Good way to start a morning with math .
Good morning!
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 🤔
He lived in Prague, but his native language was German and his surname is Italian, so is pronounced /ts/.
Interesting!
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
But that’s what we’ve shown. And c+epsilon is an upper bound since there’s nothing between c and c+epsilon basically
@@drpeyam I meant because c+epsilon can be greater than b. The set is only concerned with x in [a,b]. The way I understood it was since x_n for n>N, is not in (c-e, c+e), and we know c-e
Ok. I understand now. It's very subtle. Thanks.
brining me back to my real analysis course
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.
I know is something totally different but, talking about Weierstrass, what about a video talking about Stone-Weierstrass theorem? 👀
Haha maybe, but it would take an hour to write down the assumptions 😂
Thank you very much for the proof!
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 ?
No in that case C = {-1} because xn < -1 for 0 values of n, hence finitely many values of n
Good Job Sir 👍👍👍👍
Aweßome video as always!
Very nice proof
Beautiful proof thx!
Noice proof