I am having trouble understanding dedekind cuts. I know what they are and could find it if it was in a real line. but i am having difficulty understanding these 2 problems
Let $(A_1, A_2)$ be a Dedekind cut of $Q$. Prove that if $x < a$ and $a \in A_1$ then $x \in A_1$.
Prove that if $x > a$ and $a \in A_2$ then $x \in A_2$
Let $(A_1, A_2)$ and $(B_1, B_2)$ be Dedekind cuts of $Q$. Prove that either $A_1 \subseteq B_1$ or $A_1 \supseteq B_1$ .