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 $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ of Cauchy sequences of rational numbers.

Let $x = \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right], y = \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$, where $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ and $\left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]$ are such equivalence classes.

Thus:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x + y\)	\(=\)	\(\displaystyle \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right] + \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of real numbers
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left \langle {x_n + y_n} \right \rangle}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of real addition
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left \langle {y_n + x_n} \right \rangle}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Rational Addition is Commutative
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{\left \langle {y_n} \right \rangle}\right]\!\right] + \left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of real addition
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle y + x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\blacksquare$

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$