Suppose F is an ordered field that contains the rational num

Suppose F is an ordered field that contains the rational numbers Q. such that Q is dense, that is: whenever x, y epsilon F are such that x

Solution

Real numbers are the \"equivalence class\" of Cauchy sequences of rational numbers. You take the set of all Cauchy sequences of rational numbers and put an equivalence relation on it. We say two Cauchy sequences of rationales say x_n and y_n are relate if for every given >0 ( is rational here), there exists a n₀ such that n>n₀ implies

|x_ny_n|<

Now you say that each equivalence class is a real number. We see the reational numbers inside real numbers if r is a rational number, then the constant sequence x_n=r is a Cauchy sequence and the equivalence class it belongs to is the rational number r in the set of real numbers. This is the identification of rationals inside the reals.

yet another axiomatic way to think about the real numbers is that it is a complete ordered field, the natural numbers is the smallest inductive subset of real numbers, the integers are the subgroup generated by natural numbers, and the rationals are the field of fraction of integers.