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Supremums of unbounded sets

  • Thread starter v.rad
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  • #1
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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....
 

Answers and Replies

  • #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)|.
 
  • #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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