(related to Proposition: Existence of Inverse Real Numbers With Respect to Addition)
Let \(x=(x_n)_{n\in\mathbb N}+I\) be a real number. Because there exists an inverse rational Cauchy sequence \((-x_n)_{n\in\mathbb N}\), such that
\[(x_n)_{n\in\mathbb N}+(-x_n)_{n\in\mathbb N}=(0)_{n\in\mathbb N},\]
there exists the real number \(-x:=(-x_n)_{n\in\mathbb N}+I\). From the definition of adding real numbers, we get
\[\begin{array}{rcll} x+(-x)&=&((x_n)_{n\in\mathbb N}+I) + ((-x_n)_{n\in\mathbb N}+I)&\text{by definition of real numbers}\\ &=&(x_n+(-x_n))_{n\in\mathbb N}+I&\text{by definition of adding real numbers}\\ &=&(0)_{n\in\mathbb N}+I&\text{by existence of inverse rational Cauchy sequences with respect to addition}\\ &=&0&\text{by existence of real zero} \end{array}\]
where \(0\) denotes the real zero, as required.
