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Prove D U D' is bounded

  1. Apr 19, 2005 #1
    The homework question is this:
    Prove If D is a bounded subset of R then D bar = D U D’ is also bounded where D’ is the set of accumulation points of D.

    What is a general outline of a proof?
  2. jcsd
  3. Apr 20, 2005 #2
    It suffices to show that D' is bounded, as the union of 2 bounded sets is bounded.

    If D is bounded, then it is contained in some finite interval, ( - N, N ). If D' is not bounded, then we can find an element, x, of D' outside of ( -N, N ), and taking a suitable neighborhood around x ( of radius less than |x| - N ), we see that it is disjoint from D ( as it is disjoint from ( -N, N ) ). Therefore, x is not an accumulation point of D. Contradiction
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