This is all standard mathematics. The equivalence relationship is that the sequence (x_1, x_2, x_3, ...) is equivalent to (y_1, y_2, y_2, ...) if and only if t

This is all standard mathematics. The equivalence relationship is that the sequence (x_1, x_2, x_3, ...) is equivalent to (y_1, y_2, y_2, ...) if and only if the limit as n goes to infinity of x_n - y_n = 0. Formally, the real number represented by (x_1, x_2, x_3, ...) is the set of all Cauchy sequences which are equivalent to that one. Since "equivalent to" is transitive, any Cauchy sequence in that set will define the same set. Addition and multiplication are defined elementwise. Proving that they are well-defined is relatively straightforward. Their algebraic properties follow for free. Any rational number q can be mapped to the Cauchy sequence (q, q, q, ...) which leads to a unique real number that we somewhat sloppily call q again. I've left some details out, but this construction is well-understood, and is how we define the completion of a metric space.