Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem

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  • เผยแพร่เมื่อ 30 ต.ค. 2024

ความคิดเห็น • 16

  • @nicolasg.b.1728
    @nicolasg.b.1728 ปีที่แล้ว +13

    Hey self-learners, be aware of a minor typo in the example of x_n = (-1)^{n} @ 40:11 It should be k >= n in both sup and inf. Not n >= k.

    • @usererror7007
      @usererror7007 4 หลายเดือนก่อน

      Real one. Thanks

  • @banana144
    @banana144 20 วันที่ผ่านมา

    When Dr Casey writes the Bolzano-Weierstrass theorem you can feel the extra pulse of the chalk hitting the board, which gives you a sense of awe you can't find in the books. There should be a warning about the camara moves at the beginning, some individuals may find them disturbing, even dangerous.

  • @aryansudan2239
    @aryansudan2239 ปีที่แล้ว +15

    horrible camera work

    • @jackbradley4737
      @jackbradley4737 ปีที่แล้ว +1

      its not that bad, ive seen a lot worse on youtube

  • @georgeyang3228
    @georgeyang3228 ปีที่แล้ว +3

    51:19 it is not straightforward to me why the BW theorem is correct given the previous theorem.

    • @miyuraining
      @miyuraining ปีที่แล้ว +1

      This theorem is proven when a convergent subsequence is given. Such a subsequence consists of Xn which is sup{Xk:k>=n}. We have proved this subsequence is monotone and bounded, thus it's convergent by our previous theorem.

  • @aryansudan2239
    @aryansudan2239 ปีที่แล้ว +2

    damn good explanation

  • @YiliangWang
    @YiliangWang ปีที่แล้ว

    Bravo!

  • @MostInterestingChannel
    @MostInterestingChannel 7 หลายเดือนก่อน

    this is nice

  • @strongsim4774
    @strongsim4774 ปีที่แล้ว +2

    damn cool~~

  • @gabbiewolf1121
    @gabbiewolf1121 11 หลายเดือนก่อน +1

    1:02:40 Alternatively instead of defining an n_0 is one can define the sequence of indices for the lower and upper bounding sequences as
    n'_k =
    {1; k = 1
    {n_(k-1)+1; k>1
    That way you simply have
    a_(n'_k) - 1/k < x_(n_k)

  • @mohit4679
    @mohit4679 2 ปีที่แล้ว +2

    56:00

  • @thomasparker4109
    @thomasparker4109 2 ปีที่แล้ว +1

    cool

  • @coeusmaze9413
    @coeusmaze9413 ปีที่แล้ว +4

    Hi, I would like a sandwich with x

  • @gabbiewolf1121
    @gabbiewolf1121 11 หลายเดือนก่อน +1

    17:55 There's a problem with the proof here. x_n < sqrt(2/[n-1]) only holds for n > 1. For n = 1 the right hand side is undefined. You would have to replace the right hand side with something like
    a_n =
    {2; n 1