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

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

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