# Proof

Let $$x$$ and $$y$$ be real numbers, which by definition means that they are the equivalence classes $\begin{array}{rcl}x&:=&(x_n)_{n\in\mathbb N} + I,\\ y&:=&(y_n)_{n\in\mathbb N} + I.\end{array}$ In the above definition, $$(x_n)_{n\in\mathbb N}$$ and $$(y_n)_{n\in\mathbb N}$$ denote elements of the set $$M$$ of all rational Cauchy sequences, which represent the real numbers $$x$$ and $$y$$, while $$I$$ denotes the set of all rational sequences, which converge to $$0$$.

The commutativity of the addition of real numbers $$x+y=y+x$$ for all $$x,y\in\mathbb R$$ follows from the commutativity of adding rational Cauchy sequences. For all rational Cauchy sequences $$(x_n)_{n\in\mathbb N}, (y_n)_{n\in\mathbb N}\in M$$ we have $\begin{array}{rcll} x+y&=&((x_n)_{n\in\mathbb N}+ I)+((y_n)_{n\in\mathbb N}+ I)&\text{by definition of real numbers}\\ &=&[(x_n)_{n\in\mathbb N}+ (y_n)_{n\in\mathbb N}]+ I&\text{by definition of adding real numbers}\\ &=&[(y_n)_{n\in\mathbb N}+(x_n)_{n\in\mathbb N}]+ I&\text{by commutativity of adding rational Cauchy sequences}\\ &=&((y_n)_{n\in\mathbb N}+ I)+((x_n)_{n\in\mathbb N}+ I)&\text{by definition of adding real numbers}\\ &=&y+x&\text{by definition of real numbers}\\ \end{array}$

Github: ### References

#### Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013