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The least upper bound property and the irrationals.

  1. Aug 18, 2012 #1
    Hi

    Does anybody know if the irrational numbers have the least upper bound property?
     
  2. jcsd
  3. Aug 18, 2012 #2
    The set of irrationals less than zero is nonempty (it contains -pi, for example) and is bounded above (by pi, for example) yet has no least upper bound. So the irrationals do not satisfy the LUB property.

    Of course zero is a LUB for that set in the reals, but 0 is not irrational. That's the beauty of the LUB concept. It encapsulates the intuition of there being "no holes" in a given set.
     
  4. Aug 18, 2012 #3
    Cool that makes perfect sense thanks for the help.
     
  5. Aug 19, 2012 #4

    Bacle2

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    To generalize SteveL's answer, for any rational q,

    consider the intervals (-oo,q) . Notice that the LUB of a subset of real numbers is

    a limit point of that set S . So if a subset S of R does not contain all its limit points you

    can constructuct a subset of S that does not contain its LUB-- so that closed subsets

    contain their LUB's. Think of the relation with completeness of a set...
     
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