Real numbers, the construction from Cauchy Sequences

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