## 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:

 $\displaystyle x + y$ $=$ $\displaystyle \eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {}$ Definition of Real Number $\displaystyle$ $=$ $\displaystyle \eqclass {\sequence {x_n + y_n} } {}$ Definition of Real Addition $\displaystyle$ $=$ $\displaystyle \eqclass {\sequence {y_n + x_n} } {}$ Rational Addition is Commutative $\displaystyle$ $=$ $\displaystyle \eqclass {\sequence {y_n} } {} + \eqclass {\sequence {x_n} } {}$ Definition of Real Addition $\displaystyle$ $=$ $\displaystyle y + x$ Definition of Real Number

$\blacksquare$