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Supremums of unbounded sets

  1. Aug 29, 2010 #1
    1. sup (empty set) = -infinity, and if V is not bounded above, then sup V = +infinity. Prove if V[tex]\subseteq[/tex]W[tex]\subseteq[/tex]Real Numbers then sup V is lessthan/equalto supW




    3. I used a proof by contrapositive, but I'm not sure if it is completely valid....
     
  2. jcsd
  3. Aug 29, 2010 #2

    hunt_mat

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    I am assuming that these are intevals. You know that the lemma is clear if V=W, don't you? So assume that
    [tex]
    V\subset W
    [\tex]
    So there is an element w in W which is not an element of V, examine |w-sup(V)|.
     
  4. Aug 30, 2010 #3

    HallsofIvy

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    No reason to assume these are intervals.

    Yes, the contrapositive is the way to go. Suppose sup(W)> sup(V). Then there exist x such that sup(V)< x< sup(W). From that it follows that there exist a member of W larger than x and so larger than any member of V, a contradiction.
     
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