Real Addition is Commutative

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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

$\blacksquare$


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