The field R is a complete archimedian ordered field What doe

The field R is a complete archimedian ordered field. What does it mean to be \"complete\"?

Solution

\"Every non empty set of real numbers that has an upper bound also has a supremum in R.\"

This property of R is called The Completeness Property of R and due to this property only R is called a Complete.

This property can also be stated for lower bound and infimum as \"Every non empty set of real numbers that has a lower bound also has an infimum in R.\"

This is a deep and fundamental property of the real number system, which is frequently used in the study of Real Analysis.