A.PRIYA

  Construction of the real numbers [ edit ] Main article:  Construction of the real numbers The theorems of real analysis rely on the properties of the  real number  system, which must be established. The real number system consists of an  uncountable set  ( � ), together with two  binary operations  denoted  +  and  ⋅ , and an  order  denoted  < . The operations make the real numbers a  field , and, along with the order, an  ordered field . The real number system is the unique  complete  ordered field , in the sense that any other complete ordered field is  isomorphic  to it. Intuitively, completeness means that there are no 'gaps' in the real numbers. This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers  � ) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often...