Least Upper Bound Property
The least-upper-bound property states that every nonempty set of real numbers having an upper bound must have a least upper bound (or supremum).
The rational number line Q does not have the least upper bound property. An example is the subset of rational numbers
The number 5 is certainly an upper bound for the set. However, this set has no least upper bound in Q: the least upper bound as a subset of the reals would be, but it does not exist in Q . For any upper bound x ∈ Q, there is another upper bound y ∈ Q with y < x.
The least upper bound property can be generalized to the setting of partially ordered sets. See completeness (order theory).
Read more about this topic: Completeness Of The Real Numbers, Forms of Completeness
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