The Bolzano Weierstrass Theorem

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We prove the Bolzano-Weierstrass Theorem. This asserts that any bounded sequence of real numbers contains a convergent subsequence.

#mikedabkowski, #mikethemathematician, #profdabkowski, #realanalysis
  1. Mike, the Mathematician
  2. Bolzano-Weierstrass
  3. mike the mathematician
  4. mike dabkowski math
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Комментарии
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I have attempted to understand the Bolzano-Weierstrass Theorem multiple times, but I was still confused until I watched your explanation. I am truly grateful for your video. Thank you so much!

Music-bbrw
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The one that I learned at my uni was on the xy-plane.
I didn’t know that this theorem is also applied to 1 dimension.

akiraN
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Do you have a proof of the nested interval property?

aubrey
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Wow, are you writing everything in mirror-image?? great explanation

lucvansprang
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So basically the theorem states that in a bounded sequence of real numbers, there must exist a converging subsequence? I saw another proof where a person just kept cutting the interval into halves and picking out a random a_n (the video from Zach Star). He kept talking about how the order is important. Could you perhaps explain that to me or make an example?

Brandonhuynh
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I still don't understand why one of those intervals "must" contains infinitely many a_n
Is it obviously true or based on some statements ?

tungle
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