Given a rational number produce a larger rational number

Question

Given Dedekind cuts $A|B$ and $C|D$ in $\mathbb{Q}$, let $E=\{a+c:a\in A,c\in C \}$ Prove that E has no largest element.

If I understand the first statement $A|B$ and $C|D$ are real numbers, but $A\subseteq\mathbb{Q}$ and $C\subseteq\mathbb{Q}$. So $E\subseteq\mathbb{Q}$.

So if I let $e\in E$ then how can I produce a larger rational number?

Answer 1

Since $(A,B)$ and $(C,D)$ are Dedekind cuts, $A$ and $C$ have no greatest element.

Let $e_0 \in E$. Then $e_0 = a_0 + c_0$ for some $a_0 \in A$ and $c_0 \in C$.

Since $A$ has no greatest element, there is $a_1 \in A$ such that $a_1 > a_0$.

Then there is $e_1 \in E$ such that $e_1 = a_1 + c_0 > a_0 + c_0 = e_0$.

That means, for every element in $E$, there is a bigger element in $E$. Therefore $E$ contains no largest element.