Axiomatic Definition

Real analysis is the study of the set of Real numbers. We define the real numbers to be a set such that the following properties hold.

Axiomatic definition

Definition

The real numbers are a set denoted by \mathbb{R}, such that the following properties hold

(R1) \mathbb{R} is a field under addition and multiplication.

(R2) \mathbb{R} is totally ordered for \leq.

(R3) Addition and multiplication preserve order.

(R4) Every non-empty subset of \mathbb{R} that is bounded above has a least upper bound.

This definition is more of a theorem as we have yet to prove that any set, let alone a unique set, satisfy these properties.

Construction

TO-DO: Dedekind Cut Construction