Least-upper-bound property

The least-upper-bound property is an example of the aforementioned completeness properties which is typical for the set of real numbers.

If an ordered set S has the property that every nonempty subset of S having an upper bound also has a least upper bound, then S is said to have the least-upper-bound property. As noted above, the set R of all real numbers has the least-upper-bound property. Similarly, the set Z of integers has the least-upper-bound property; if S is a nonempty subset of Z and there is some number n such that every element s of S is less than or equal to n, then there is a least upper bound u for S, an integer that is an upper bound for S and is less than or equal to every other upper bound for S.

An example of a set that lacks the least-upper-bound property is Q, the set of rational numbers. Let S be the set of all rational numbers q such that q2 < 2. Then S has an upper bound (1000, for example, or 6) but no least upper bound in Q. For suppose p ∈ Q is an upper bound for S, so p2 > 2. Then q = (2p+2)/(p + 2) is also an upper bound for S, and q < p. (To see this, note that q = p − (p2 − 2)/(p + 2), and that p2 − 2 is positive.)

There is a corresponding 'greatest-lower-bound property'; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least-upper-bound of the set.

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