# Proof

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

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 $$0$$.

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

Because the addition of rational Cauchy sequences is cancellative, it follows $\begin{array}{rcll} x+z=y+z&\Leftrightarrow&(x_n+z_n)_{n\in\mathbb N}+I=(y_n+z_n)_{n\in\mathbb N}+I&\text{by definition of adding real numbers}\\ &\Leftrightarrow& (x_n)_{n\in\mathbb N}+I=(y_n)_{n\in\mathbb N}+I&\text{because addition of rational Cauchy sequences is cancellative}\\ &\Leftrightarrow& x=y&\text{by definition of real numbers} \end{array}$

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

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