Proof

Because the multiplication of real numbers is commutative, it is without loss of generality sufficient to show the right cancellation property, i.e. $x\cdot z=y\cdot z\Leftrightarrow x=y,~~~~~~(x,y,z\in\mathbb R,~z\neq 0).$

By definition of real numbers, the numbers $$x,y,z$$ are some equivalence classes $\begin{array}{rcl}x&:=&(x_n)_{n\in\mathbb N}+I,\\y&:=&(y_n)_{n\in\mathbb N}+I,\\z&:=&(z_n)_{n\in\mathbb N}+I\\\end{array}$

for some rational Cauchy sequences $$(x_n)_{n\in\mathbb N}$$, $$(y_n)_{n\in\mathbb N}$$, 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 $$0$$. Moreover, since $$z\neq 0$$ by hypothesis, the rational Cauchy sequence $$(z_n)_{n\in\mathbb N}$$ is not convergent to 0, i.e. $$(z_n)_{n\in\mathbb N}\not\in I$$.

By definition of multiplying real numbers, we have $\begin{array}{rcl} x\cdot z&=&(x_n\cdot z_n)_{n\in\mathbb N}+I,\\ y\cdot z&=&(y_n\cdot z_n)_{n\in\mathbb N}+I. \end{array}$

By hypothesis, $$(z_n)_{n\in\mathbb N}\not\in I$$. Therefore, there exist $$N\in\mathbb N$$ such that $$|z_n|>0$$ for all $$n > N$$. For such indices it follows from the cancellation property of multiplying rational Cauchy sequences that

$\begin{array}{rcll} x\cdot z=y\cdot z&\Leftrightarrow&(x_n\cdot z_n)_{n > N}+I=(y_n\cdot z_n)_{n > N}+I&\text{by definition of multiplying real numbers}\\ &\Leftrightarrow& (x_n)_{n > N}+I=(y_n)_{n > N}+I&\text{because multiplication of rational Cauchy sequences is cancellative}\\ &\Leftrightarrow& x=y&\text{by definition of real numbers} \end{array}$

Altogether, we have shown that the addition of real numbers is cancellative $x\cdot z=y\cdot z\Rightarrow x=y,~~~~~~(x,y,z\in\mathbb R,~z\neq 0),$ and its conversion $x=y\Rightarrow x\cdot z=y\cdot z,~~~~~~(x,y,z\in\mathbb R,~z\neq 0).$

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References

Bibliography

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