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zepdrix

  • one year ago

(Advanced Calculus I) Problem to follow.

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  1. zepdrix
    • one year ago
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    Suppose that the real number \(\large\rm a\) has the property that for every natural number \(\large\rm n\), \(\large\rm a\le\dfrac{1}{n}\). Prove that \(\large\rm a\le0\).

  2. zepdrix
    • one year ago
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    I'm thinking I need to use the Archimedean Property in some way here... struggling with it though... hmm

  3. sohailiftikhar
    • one year ago
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    how it can be true ok take n=5 so 1/5 is not smaller than zero right ?

  4. zepdrix
    • one year ago
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    If you look at all the fractions on the right side of the inequality, the larger and larger than n gets, the smaller and smaller the fraction gets overall. So it's getting really really close to zero. So a, which is smaller than those set of values has to be zero or below since the fractions take up all of the positives. Conceptually I think I understand. I'm just having trouble proving it XD

  5. sohailiftikhar
    • one year ago
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    oh! yes you can use your word to prove it use them into language of math . just make two equations and draw the conclusion .

  6. zepdrix
    • one year ago
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    Do I need to do something with the Completeness Axiom? :o Like if I can show that 0 is the infimum of the set of these (1/n)'s, does that get me on the right track? @zzr0ck3r @ganeshie8 Gotta get the big guns up in hereee XD

  7. zzr0ck3r
    • one year ago
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    This is Archimedes all the way

  8. zzr0ck3r
    • one year ago
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    Suppose \(a>0\), then there exists \(n\) s.t. \(\dfrac{1}{n}<a\) and we are done.

  9. anonymous
    • one year ago
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    lol @zzr0ck3r yeah in fact this is the property itself :)

  10. sohailiftikhar
    • one year ago
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    oh yes just suppose a>0 is correct and you will find that this is not correct then a<0 will be correct that's all.

  11. zzr0ck3r
    • one year ago
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    \(a\le 0\)

  12. zepdrix
    • one year ago
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    Suppose a>0, then there exists n s.t. 1/n<a and we are done. Hmm I'm confused... how is that `not correct` though? :o

  13. zzr0ck3r
    • one year ago
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    If you try and use some inf argument, you will end up using the Archimedian property anyway ...I think

  14. zzr0ck3r
    • one year ago
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    Because we were told that \(a<\frac{1}{n}\) for all \(n\).

  15. zepdrix
    • one year ago
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    Oh oh oh great stuff! :) For some reason I thought we were contradicting the fact that a>0. Ok ok ok that makes more sense.

  16. anonymous
    • one year ago
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    assume a>0, then there exist \(s\in N\) s.t a>1/s , which is a contradiction

  17. zzr0ck3r
    • one year ago
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    yeah, it seems awful circular but it isn't :)

  18. zepdrix
    • one year ago
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    Halmos what is that picture? +_+ What is she so exciting about..? Is that... pie?

  19. anonymous
    • one year ago
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    i felt so but maybe its away to look at it and understand, but i wonder this is not even calculus this is real analysis don't you feel like ur insulting pure math by saying calculus. @zepdrix yeah it is :P i like pies lol

  20. zepdrix
    • one year ago
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    Real Analysis? 0_o I've been lied to? Gasp...

  21. anonymous
    • one year ago
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    well idk lol it might be just an introduction, im just saying xD

  22. zepdrix
    • one year ago
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    Ya, the book is titled "Advanced Calculus, second edition". They're probably just taking it nice and easy this first two weeks of class :)

  23. zzr0ck3r
    • one year ago
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    it is all calculus. what you call calculus some people call baby math. I know profs that refer to 600 level analysis classes as calculus

  24. zzr0ck3r
    • one year ago
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    I would not call this real analysis. I would call it advanced calculus.

  25. anonymous
    • one year ago
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    hmm

  26. zzr0ck3r
    • one year ago
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    My defining line is: when it gets hard :)

  27. zzr0ck3r
    • one year ago
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    even \(\epsilon-delta\) continuity/limit proofs I call advanced calculus

  28. zepdrix
    • one year ago
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    I'm taking Abstract Algebra I at the same time as this class. That class has me very excited. Such interesting stuff :OOOO

  29. zzr0ck3r
    • one year ago
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    good for you. I much prefer abstract algebra.

  30. anonymous
    • one year ago
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    abstract algebra is my favorite topic :D

  31. zzr0ck3r
    • one year ago
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    same, and algebraic topology is the sh*t

  32. anonymous
    • one year ago
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    geometric topology existed me :3 idk im in love with pure math and sucks in applied math xD

  33. zepdrix
    • one year ago
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    @Halmos it brought you into existence? 0_o

  34. zzr0ck3r
    • one year ago
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    I have taken general topology and algebraic topology. I start computational topology in the fall. That should either be really cool or super lame.

  35. zzr0ck3r
    • one year ago
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    :)

  36. anonymous
    • one year ago
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    hmmm typo excited me xD

  37. zzr0ck3r
    • one year ago
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    three people that hate applied math and love pure math excites me:)

  38. anonymous
    • one year ago
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    awesome <3

  39. zzr0ck3r
    • one year ago
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    I got pretty liberal with the word hate.

  40. zepdrix
    • one year ago
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    Ehhh I still love differential equations :) lol

  41. zepdrix
    • one year ago
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    And anything with complex numbers gets me all giddy XD

  42. zzr0ck3r
    • one year ago
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    Then you will like analysis once you get used to it. Booooo

  43. zzr0ck3r
    • one year ago
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    jk, I like it as well. I T.A. for that class. I really just hate grading the HW. Group theory HW is fun to grade.

  44. zzr0ck3r
    • one year ago
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    I ta analysis that is...

  45. zepdrix
    • one year ago
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    oh neato :O

  46. anonymous
    • one year ago
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    i like geometry also xD

  47. zzr0ck3r
    • one year ago
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    not difeq, I suck at dif eq. But I once had my class rolling in a difeq class. The teacher was trying to explain something and he said "imagine you had a perfect carpenter" I yelled out JESUS!

  48. anonymous
    • one year ago
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    wth xD

  49. zzr0ck3r
    • one year ago
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    I have worked through some of Elements(100 pages or so), but that is about all I know in geometry...

  50. zzr0ck3r
    • one year ago
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    ok back to more fun questions for us @zepdrix

  51. zepdrix
    • one year ago
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    \c:

  52. anonymous
    • one year ago
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    well Euclid, Hilbert, spherical,hyperbolic blah blah all kind of geometry i love <3 and @zepdrix yeah share with us :D

  53. zzr0ck3r
    • one year ago
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    Hilbert had his own geometry also. Damn!!! he did that late in the game...

  54. zzr0ck3r
    • one year ago
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    is it just a bunch of hotels in two space? jk

  55. anonymous
    • one year ago
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    well in fact he modified Euclid elements and make it more neat and added most of logical thoughts of it, but its known for most its euclidean only :P

  56. zzr0ck3r
    • one year ago
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    known is a big word :)

  57. zzr0ck3r
    • one year ago
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    p.s. @zepdrix showing that the inf is 0 is a great idea and in general that is what you will be doing, but I don't think you can without the Arch property in this case, or something that needs the arch prop in order to prove it. ..

  58. zzr0ck3r
    • one year ago
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    Also, this could be stronger, change your statement to Suppose that the real number \(a\) has the property that for every natural number \(n\), \(a<\frac{1}{n}\).

  59. Mimi_x3
    • one year ago
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    what math course out there is not hard????

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