# The least upper bound property and the irrationals.

Hi

Does anybody know if the irrational numbers have the least upper bound property?

Hi

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.

Bacle2