Real Addition is Commutative

Theorem
The operation of addition on the set of real numbers $\R$ is commutative:


 * $\forall x, y \in \R: x + y = y + x$

Proof
From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers.

Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equivalence classes.

Thus: