Real Analysis | The limit point of a set A⊆ℝ

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  • เผยแพร่เมื่อ 22 ธ.ค. 2024

ความคิดเห็น • 50

  • @tomatrix7525
    @tomatrix7525 3 ปีที่แล้ว +4

    These videos are absolutely awesome. You’ve no idea how many people you actually teach Michael, you are basically my teacher

  • @arvindsrinivasan424
    @arvindsrinivasan424 4 ปีที่แล้ว +41

    This reminds me of a topology class I shouldn't have passed

  • @sarojpandeya7883
    @sarojpandeya7883 4 ปีที่แล้ว +8

    Thanks Dr Penn for making amazing videos on mathematical analysis

  • @virat.chauhan
    @virat.chauhan 3 ปีที่แล้ว

    He is definitely the best math teacher at TH-cam!

  • @manniman82
    @manniman82 ปีที่แล้ว

    @Michael Penn Are the limit points of a sequence the same thing as a limit point of a set? I am using Terrence Tao's Analysis 1 and 2 and he defined the limit point of a sequence in a metric space (X,d) as follows:
    ''Suppose that x_n (from n=m to n=infinity) is a sequence of points in a metric space (X,d). Let L be in space X. We say L is a limit point of the sequence x_n if and only if for every N>=m and for every (epsilon) > 0 there exists an n>=N such that d(x_n, L) =n} (as I did in step 2) is not correct? Or are the definitions for limit points of sequence and limit points of sets not the same thing?

  • @elasmarsaadallah6126
    @elasmarsaadallah6126 4 ปีที่แล้ว +1

    6:42 we can also prove that 0 is the only limit point of A ={1/n, n ≥ 1}

    • @elasmarsaadallah6126
      @elasmarsaadallah6126 4 ปีที่แล้ว

      indeed for all aₓ = 1/x we can take any £ < d(1/x, 1/x+1)= 1/(x²+x), for exemple d/2

  • @azhakabad4229
    @azhakabad4229 4 ปีที่แล้ว +9

    Sir, when will you start linear algebra ?

  • @goodplacetostop2973
    @goodplacetostop2973 4 ปีที่แล้ว +12

    16:11

  • @AQrdmbmaster
    @AQrdmbmaster 4 ปีที่แล้ว +8

    I’m wondering if you would give some motivation regarding the usefulness of these theorems? Even if you merely say “this helps us prove xyz in the future.” Basically, tell us why these concepts matter in the larger picture.
    Aside: I’m absolutely loving your real analysis series. I look forward to more episodes!

    • @yuklungleung620
      @yuklungleung620 4 ปีที่แล้ว +1

      They are just so basic just like 1+1=2

    • @schweinmachtbree1013
      @schweinmachtbree1013 4 ปีที่แล้ว +3

      limit points are used in order to define limits of functions, for example. in the context of "the limit of f at c is L", c is a limit point of the domain of f (by the definition of a limit point, we can "approach" c, getting arbitrarily close to it, but without actually getting *to* it, which is exactly the behaviour we want for defining limits) :)

    • @danmwas7967
      @danmwas7967 3 ปีที่แล้ว

      @@yuklungleung620 😂

  • @clonebin0
    @clonebin0 4 ปีที่แล้ว +1

    4:26 nice transition

  • @anirudhranjan7002
    @anirudhranjan7002 3 ปีที่แล้ว

    Is the second theorem correct? I found a counter example but not sure if it's correct.... the sequence a_n = Cos(npi/2) /n and the set A be { +-1/n such that n is natural number } U {0} . Clearly sequence lies inside the set A and coverges to 0 but also multiple terms of the sequence are 0. Given that the reverse condition is not satisfied, we still have that 0 is a limit point of the set A. Did i go wrong anywhere? Can someone help me out......

  • @RandomBoy_0o0
    @RandomBoy_0o0 2 ปีที่แล้ว

    Sir I think that at 3:43 you meant "2+e/2 is definitely not equal to 2..."

  • @izumiasmr
    @izumiasmr ปีที่แล้ว

    I wonder, shouldn't we take some care of the situation that some a_n's might be equal, I mean there's nothing wrong with it even according the very definition of a limit of sequence (the induced sequence), however I think we intuitively expect all the terms a_n to be distinct and I think it's good to point out in the very statement of the teorem that it's not an issue (in the end taking a finite number of a_n's won't work as a sequence converging to x, as a_n's should not be equal to x, whereas the fact that we might end up taking particular elements of the set more than once is not itself a problem)

  • @hartandrin7565
    @hartandrin7565 4 ปีที่แล้ว +1

    Do we need to invoke the axiom of choice in order to choose some a_n in the (possibly) infinite set that you've constructed in your "if" direction? Is there any proof that you know of that doesn't rely on the axiom of choice?

    • @schweinmachtbree1013
      @schweinmachtbree1013 4 ปีที่แล้ว

      just the Axiom of Countable Choice (since we are choosing a_n for each n, so we are making a countable number of choices), which is relatively weak; it doesn't have dramatic consequences like the ordinary Axiom of Choice, e.g. the Banach Tarski paradox. I am 90% certain that Countable Choice is needed, and I'm sure you can confirm that easily with a Google search

  • @parthmani3877
    @parthmani3877 3 ปีที่แล้ว +1

    Please upload a series of measure theory.

  • @sumittete2804
    @sumittete2804 2 ปีที่แล้ว

    Sir...what are the limit points of the set {cosn: n is any natural number}?

  • @lynettemojica6503
    @lynettemojica6503 3 ปีที่แล้ว

    5:30 won't we be out of the interval if epsilon is great than 2. Should we put a cap on epsilon so it stays in the interval?

  • @punditgi
    @punditgi 4 ปีที่แล้ว

    Really helpful to get a handle on point set topology of the set of real numbers. Thanks for getting this ball rolling!
    One issue that I hope you can clarify is this: Various authors I have read have lumped together limit point, accumulation point, and cluster point. However, I came across at least one author who tried to define all 3 somewhat differently but I can no longer find that reference. Are you able to draw any distinctions among those 3 terms?

    • @schweinmachtbree1013
      @schweinmachtbree1013 4 ปีที่แล้ว

      Wikipedia should help you (or the references therein if you don't like wikipedia itself)

  • @scrappybuilds
    @scrappybuilds 3 ปีที่แล้ว +1

    Thanks for all your great instructional videos! I was going to hit the like button but this video is currently at pi likes and I didn't have the heart to change that

  • @youssefamen6872
    @youssefamen6872 4 ปีที่แล้ว +1

    Pls do imo 2006 problem 5 thx for the hard work.

  • @henrywoo1668
    @henrywoo1668 4 ปีที่แล้ว

    Thank you sir for your great lecture .

  • @gregorymacchio4077
    @gregorymacchio4077 3 ปีที่แล้ว

    Your content is amazing!

  • @cicik57
    @cicik57 4 ปีที่แล้ว

    emm i may be wrong but you do not need to have epsilon >0 since the Ve ist symmetrical around zero and you need just epsilon not equal to zero

  • @hyperboloidofonesheet1036
    @hyperboloidofonesheet1036 4 ปีที่แล้ว +1

    Really grinding up the chalk today.

  • @noumaneelgaou1624
    @noumaneelgaou1624 4 ปีที่แล้ว

    Tank you techer for this vidios can you plise explained graph theory

  • @debjitmullick7004
    @debjitmullick7004 4 ปีที่แล้ว

    Sir ... Is it necessary, that limit point should be ... In side the set ....???

    • @XavierMacX
      @XavierMacX 4 ปีที่แล้ว +1

      No it does not have to be. He addressed this last video :) cheers.

  • @cah220
    @cah220 4 ปีที่แล้ว

    An interesting question (gleefully lifted from an exercise in baby Rudin) that I'd like to see a detailed solution to (been 20 years since I took undergraduate Analysis). "Construct a bounded set of real numbers with exactly 3 limit points." I get lost in a few of the details.

    • @billjoe5991
      @billjoe5991 2 ปีที่แล้ว +1

      The easiest way of solving that that I can think of off the top of my head is by finding 3 sequences that all aren’t “close” to one another and that all converge to different points. You can then just intersect those 3 sequences or smack them together as subsequences into one main sequence, and you’ll have a set with 3 different limit points. An example would be the set of points of the sequences:
      1) A= {5-1/n : n in Z}
      2) B={10 -1/n : n in Z}
      3) C= {20 + 1/n : n in Z}
      Then the set D = A U B U C has 3 limit points which are {5, 10, 20}

  • @johnalley8397
    @johnalley8397 4 ปีที่แล้ว +1

    Another great video.
    Wonder if you might be on the lookout for opportunities to deliver an informal explanation of the intuition of a thing. The proofs get a bit dry and abstract and sometimes i just need a sentence or two on the jist of a thing to orient my thinking a bit.
    When I heard the learn’d astronomer,
    When the proofs, the figures, were ranged in columns before me,
    When I was shown the charts and diagrams, to add, divide, and measure them,
    When I sitting heard the astronomer where he lectured with much applause in the lecture-room,
    How soon unaccountable I became tired and sick,
    Till rising and gliding out I wander’d off by myself,
    In the mystical moist night-air, and from time to time,
    Look’d up in perfect silence at the stars.
    - WALT WHITMAN

    • @yuklungleung620
      @yuklungleung620 4 ปีที่แล้ว +1

      Real analysis is rigorous, no informal explanation

    • @johnalley8397
      @johnalley8397 4 ปีที่แล้ว

      @@yuklungleung620 Brilliant observation. 100% true if you want to reach only the folks generally reached.
      If you want to expand the appeal of these videos to a wider audience, lead with a sentence or two about the intuition.
      3b1b is the only necessary supporting argument here.

  • @rafael7696
    @rafael7696 4 ปีที่แล้ว

    In Spain is called acumulation point

  • @birdboat5647
    @birdboat5647 4 ปีที่แล้ว +10

    this is so hot

  • @soranuareane
    @soranuareane 4 ปีที่แล้ว +3

    Maybe first? Doubt it.

    • @failsmichael2542
      @failsmichael2542 4 ปีที่แล้ว +3

      That’s a good place to start.

    • @soranuareane
      @soranuareane 4 ปีที่แล้ว +1

      @@failsmichael2542 You win today, Michael, but alas, one day, I will one-up that.

  • @numericalcode
    @numericalcode ปีที่แล้ว

    Chalk drop

  • @muuubiee
    @muuubiee 2 ปีที่แล้ว

    Isn't that definition wrong? This part confused me so much, because the definitions given made no sense, and was obviously flawed.
    It must contain x AND at least another point other than x... Not just any point which is not x (which this definition suggests), because then every single possible point is a limit point.

  • @kanewilliams3613
    @kanewilliams3613 4 ปีที่แล้ว

    he needs to apply the brakes...!!

  • @johnstroughair2816
    @johnstroughair2816 4 ปีที่แล้ว

    About time to move your merchandise off Teespring given their association with extreme rightwing issues?

    • @MichaelPennMath
      @MichaelPennMath  4 ปีที่แล้ว

      I haven't heard of this... let me look into it.

    • @johnstroughair2816
      @johnstroughair2816 4 ปีที่แล้ว

      Michael Penn Currently trending on twitter that Teespring is selling Free Kyle (Rittenhouse) T-shirts. By the way I should add I am really enjoying your videos, since my son pointed them out to me.