# Proof

Let $$x$$ and $$y$$ be real numbers, which by definition means that they are the equivalence classes $\begin{array}{rcl}x&:=&(x_n)_{n\in\mathbb N} + I,\\ y&:=&(y_n)_{n\in\mathbb N} + I.\\z&:=&(z_n)_{n\in\mathbb N} + I.\end{array}$ In the above definition, $$(x_n)_{n\in\mathbb N}$$, $$(y_n)_{n\in\mathbb N}$$, and $$(z_n)_{n\in\mathbb N}$$ 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 associativity of the multiplication of real numbers $$(x\cdot y)\cdot z=x\cdot (y\cdot z)$$ for all $$x,y,z\in\mathbb R$$ follows from the associativity of multiplying rational Cauchy sequences. For all rational Cauchy sequences $$(x_n)_{n\in\mathbb N}, (y_n)_{n\in\mathbb N}, (z_n)_{n\in\mathbb N}\in M$$ we have $\begin{array}{rcll} (x\cdot y)\cdot z&=&[((x_n)_{n\in\mathbb N}+ I)\cdot ((y_n)_{n\in\mathbb N}+ I)]\cdot ((z_n)_{n\in\mathbb N}+I)&\text{by definition of real numbers}\\ &=&[((x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N})+ I]\cdot ((z_n)_{n\in\mathbb N}+I)&\text{by definition of multiplying real numbers}\\ &=&[((x_n)_{n\in\mathbb N}\cdot (y_n)_{n\in\mathbb N})\cdot (z_n)_{n\in\mathbb N}+I]&\text{by definition of multiplying real numbers}\\ &=&[(x_n)_{n\in\mathbb N}\cdot ((y_n)_{n\in\mathbb N}\cdot (z_n)_{n\in\mathbb N})+I]&\text{by associativity of multiplying rational Cauchy sequences}\\ &=&((x_n)_{n\in\mathbb N}+ I) \cdot [(y_n)_{n\in\mathbb N})\cdot (z_n)_{n\in\mathbb N}+I]&\text{by definition of multiplying real numbers}\\ &=&((x_n)_{n\in\mathbb N}+ I) \cdot [((y_n)_{n\in\mathbb N})+ I) \cdot ((z_n)_{n\in\mathbb N}+I)]&\text{by definition of multiplying real numbers}\\ &=&x\cdot (y\cdot z)&\text{by definition of real numbers}\\ \end{array}$

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### References

#### Bibliography

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