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Homework Help: Proof on boundedness of sets in n space

  1. May 11, 2010 #1
    1. The problem statement, all variables and given/known data
    Let S be a bounded set in n -space. Fix a d>0. Then it is possible to choose a finite set of points {pi....pm} in S such that every point p in S is within a distance d of at least one of the points p1, p2,....pm.


    2. Relevant equations

    None really.

    3. The attempt at a solution

    I've tried some methods but I have been stuck at some point of every attempt. A nudge (or two) in the right direction would be greatly appreciated.
     
    Last edited: May 12, 2010
  2. jcsd
  3. May 12, 2010 #2

    lanedance

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    is that exactly how the question is written & is there any additional info?
     
  4. May 12, 2010 #3
    This is exactly how its written and thats all they gave.
     
  5. May 12, 2010 #4

    Mark44

    Staff: Mentor

    Show us what you've tried. At the very least you should be using the definition of a bounded set in Rn.
     
  6. May 12, 2010 #5

    lanedance

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    as a start you could consider 2 cases, where the bound on S is less than d and when it is greater....
     
  7. May 12, 2010 #6
    How much point-set topology do you know? I've got a proof, but it uses a property of compact spaces that you may not be expected to know.
     
  8. May 12, 2010 #7
    What property of compactness did you need? Here is an idea that might nudge you in the right direction. Consider the special case of R, which is one-dimensional. Since S is bounded, it lies entirely between -M and M for some M. Certainly, if d >= 2M, we need just pick any point in S to be done. Every other point is within a distance 2M from that point. What if d = M? A first thought might be to choose the midpoint... but that might not be in S. How can we compensate?
     
  9. May 13, 2010 #8
    Basically just the definition: Any open cover of a compact set has a finite subcover. To get a compact set, take the closure of S; to get an open cover, consider the union of all balls of radius d around every point in S...
     
  10. May 14, 2010 #9
    That is pretty slick. Occasionally I have these bouts where I realize I'm in out of my depth and don't know enough in this case about metric spaces. I know that in Rn compactness isn't needed (at least not directly). But now I'm not so sure about arbitrary metric spaces.
     
  11. May 14, 2010 #10

    Dick

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    It's not true for an arbitrary metric space. Think about the unit sphere in an infinite dimensional Hilbert space, or even simpler any space with an infinite number of elements with a discrete metric.
     
  12. May 18, 2010 #11
    I don't know much topology, only the basic concepts and definitions such as limit point, open closed sets, boundary, closure, and such. The class is a multivariable analysis class, so only basic knowledge has been needed so far.

    I actually asked my TA the question and the proof he gave was very long and something that didn't even look undergraduate level to me. If you've found a proof using basic concepts I would appreciate it greatly if you would post it. I haven't really been able to dent the problem.

    I'm not sure. I thought about the problem a lot and I was completely convinced that the set S need be both bounded and closed for the rest to follow, so its limit points would be contained in the set.
     
  13. May 18, 2010 #12

    Dick

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    If the set in contained in Euclidean n-space, then there is a way to prove it that doesn't require any topology at all. Practice by supposing n=2. Hint: for any bounded set you can pick a number S such that the set in contained in a square with side length S centered at the origin. Now pick a d>0 and describe a finite set of points that works.
     
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