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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.
