Hey guys I have a quiz soon and I really dont know how to prove this question. I tried my best but it is not working. Please help out with anything or hints.
Let $(A'_1, A'_2)$ be a Dedekind cut of ${\mathbb{Q}}$ that represents the same real number as $(A_1, A_2)$. Let $C'_1 = A'_1 + B_1$ and $C'_2 = {\mathbb{Q}} \diagdown C'_1$.
Prove that $(C'_1, C'_2)$ represents the same real number as $(C_1, C_2)$.
By Dedekind cut, it means a pair $(A,B)$ such that:
(i) $A,B\neq\emptyset,$
(ii) $A\cap B=\emptyset,$
(iii) $A\cup B=\Bbb Q,$
(iv) $y\in A$ whenever $y\in\Bbb Q$ and there is some $x\in A$ with $y<x,$ and
(v) $A$ has no greatest element.