A real number is defined to be a new type of object, written as $$LIM_{n \rightarrow \infty} a_n$$. This object has a (1 to many) correspondence to a Cauchy sequence $$(a_n)_{n=1}^{\infty}$$. The Cauchy sequence is used to define an equivalence relation between real numbers: two real numbers $$LIM_{n \rightarrow \infty} a_n$$ and $$LIM_{n \rightarrow \infty} b_n$$ are said to be equal iff the corresponding sequences $$(a_n)_{n=1}^{\infty}$$ and $$(b_n)_{n=1}^{\infty}$$ are [...]