Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The least upper bound property and the irrationals.

  1. Aug 18, 2012 #1

    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


    User Avatar
    Science Advisor

    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...
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook