The set of real numbers can be constructed from the set of rational numbers. It’s premised from the idea that we can approximate a real number from a rational number.
We define a real number as an element of the collection of Cauchy sequences of rational numbers . The denotes an equivalence class, i.e., if the Cauchy sequences are equivalent if the differences between their terms becomes arbitrarily small.
i.e., a real number can be identified with an equivalence class of Cauchy sequences.
We consider all Cauchy sequences that get arbitrarily close to the same value. Each real number is represented by a corresponding equivalence class.