The least upper bound property and the irrationals.

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.

 

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...