Does anybody know if the irrational numbers have the least upper bound property?
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.
Cool that makes perfect sense thanks for the help.
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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