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?