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→What is the problem? Aren't rational numbers good enough?: rm untrue statement--the point of supremum of S is that it doesn't have to be in S (like 6 in open interval (3,6))
:Definition: A number L is a ''least upper bound'' (often abbreviated "lub") if it is an upper bound and no other upper bound is smaller. (There is also the notion of a greatest lower bound, abbreviated "glb".) 6 is the lub of the open interval <math>(3, 6)\,</math>. 3 is its glb. 6 and 3 are also the lub and glb of the closed interval <math>[3, 6]\,</math>—the inclusion of the endpoints makes no difference.
:Definition: A number L is a ''least upper bound'' (often abbreviated "lub") if it is an upper bound and no other upper bound is smaller. (There is also the notion of a greatest lower bound, abbreviated "glb".) 6 is the lub of the open interval <math>(3, 6)\,</math>. 3 is its glb. 6 and 3 are also the lub and glb of the closed interval <math>[3, 6]\,</math>—the inclusion of the endpoints makes no difference.
::The least upper bound is also sometimes called the "supremum", abbreviated "sup". The greatest lower bound is also sometimes called the "infimum", abbreviated "inf". For simple cases like the real numbers, the terms "maximum" and "minimum" may also be used.
::The least upper bound is also sometimes called the "supremum", abbreviated "sup". The greatest lower bound is also sometimes called the "infimum", abbreviated "inf".
A set has the ''least upper bound property'' if every set that has an upper bound has a least upper bound. There is also a ''greatest lower bound property'', and any reasonable set having one property has the other.
A set has the ''least upper bound property'' if every set that has an upper bound has a least upper bound. There is also a ''greatest lower bound property'', and any reasonable set having one property has the other.