Editing Real analysis (section)

===Construction of the real numbers===
{{Main|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]] (<math>\mathbb{R}</math>), together with two [[binary operation]]s denoted {{math|+}} and {{math|⋅}}, and a [[total order]] denoted {{math|≤}}. The operations make the real numbers a [[Field (mathematics)|field]], and, along with the order, an [[ordered field]].  The real number system is the unique ''[[Completeness (order theory)|complete]] ordered field'', in the sense that any other complete ordered field is [[Isomorphism|isomorphic]] to it. Intuitively, completeness means that there are no 'gaps' (or 'holes') in the real numbers. This property distinguishes the real numbers from other ordered fields (e.g., the rational numbers <math>\mathbb{Q}</math>) and is critical to the proof of several key properties of functions of the real numbers. The completeness of the reals is often conveniently expressed as the ''least upper bound property'' (see below).