# 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 addition of real numbers $$(x+y)+z=x+(y+z)$$ for all $$x,y,z\in\mathbb R$$ follows from the associativity of adding 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 $\begin{array}{rcll} (x+y)+z&=&[((x_n)_{n\in\mathbb N}+ I)+((y_n)_{n\in\mathbb N}+ I)]+((z_n)_{n\in\mathbb N}+I)&\text{by definition of real numbers}\\ &=&[((x_n)_{n\in\mathbb N}+ (y_n)_{n\in\mathbb N})+ I]+((z_n)_{n\in\mathbb N}+I)&\text{by definition of adding real numbers}\\ &=&[((x_n)_{n\in\mathbb N}+ (y_n)_{n\in\mathbb N})+ (z_n)_{n\in\mathbb N}+I]&\text{by definition of adding real numbers}\\ &=&[(x_n)_{n\in\mathbb N}+ ((y_n)_{n\in\mathbb N}+ (z_n)_{n\in\mathbb N})+I]&\text{by associativity of adding rational Cauchy sequences}\\ &=&((x_n)_{n\in\mathbb N}+ I) + [(y_n)_{n\in\mathbb N})+ (z_n)_{n\in\mathbb N}+I]&\text{by definition of adding real numbers}\\ &=&((x_n)_{n\in\mathbb N}+ I) + [((y_n)_{n\in\mathbb N})+ I) +((z_n)_{n\in\mathbb N}+I)]&\text{by definition of adding real numbers}\\ &=&x+(y+z)&\text{by definition of real numbers}\\ \end{array}$

Github: ### References

#### Bibliography

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