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Homework Help: How to prove something is closed and bounded, ie compact

  1. Sep 25, 2012 #1
    1. The problem statement, all variables and given/known data
    I need to prove that a closed ball(radius r about x0) is closed and bounded. The same goes for a sphere(radius r about x0).

    2. Relevant equations

    3. The attempt at a solution
    How does one go about proving something is closed and bounded? My book is not very helpful and searching hasn't yielded much. This is only a part of the problem, but the rest should be doable once I get this.
  2. jcsd
  3. Sep 25, 2012 #2
    Start by giving the definitions of "closed ball", "sphere", "closed" and "bounded".
  4. Sep 25, 2012 #3
    Well I'm lacking on the definition of "closed" and "bounded" which is what I need to get started
  5. Sep 25, 2012 #4
    What book are you using? Aren't those things in there?
  6. Sep 25, 2012 #5
    Ok so I assume i can prove something is bounded if every neighborhood of point along the edge(ie at r from x0) contains both a point in the ball and outside the ball. I just took that from the boundary point definition, now that doesn't imply something is closed does it?
  7. Sep 25, 2012 #6


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    Science Advisor
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    That's not what bounded means. In fact, bounded has nothing to do with boundary. A bounded set is one that is fits inside a ball of some finite radius.

    There are various equivalent definitions of closed. You must find out which one your book is using. Some possibilities:
    * a set is closed if and only if its complement is open
    * a set is closed if and only if it contains all of its limit points
    * a set is closed if and only if it contains all of its boundary points
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