# Dedekind cuts help

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 $x \in A_2$

Also

Let $(A_1, A_2)$ and $(B_1, B_2)$ be Dedekind cuts of $Q$. Prove that either $A_1 \subseteq B_1$ or $A_1 \supseteq B_1$ .

Assume $x\notin A_1$. Then $x\in A_2$ because $A_1\cup A_2=\mathbb Q$. Then $a<x$ because $a\in A_1$ and $x\in A_2$. This contradicts $x<a$, hence x\in A_1$. The second question works the same way. Assume neither$A_1\subseteq B_1$nor$A_1\supseteq B_1$. Then there exists$x\in A_1\setminus B_1$and$a\in B_1\setminus A_1$. If$x<a$then$a\in B_1$implies$x\in B_1$by the previous result - contradiction. If$x>a$, then$x\in A_1$implies$a\in A_1$by the previous result - contradiction. Therefor$x=a\$, even more contradiction.