(related to Proposition: Addition of Real Numbers Is Cancellative)
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 \[x=y\Rightarrow x+z=y+z,~~~~~~(x,y,z\in\mathbb R).\]
