I am confused about the proof of following claim:
Let A and B be dedekind cuts. How can we show that A is subset of B (including equality possibility) $\Longrightarrow$ $A \cup B = B$? and similarly A is subset of B (including equality possibility) $\Longrightarrow$ $A \cap B = A$?
Dedekind Cuts are just ordinary sets (or pairs of sets, depending on your notation). So, here $A,\ B$ are just two sets. Now, $ A \subset B$ means that any element that is contained in $A$ is also contained in $B$, and so 'union-ing' $A$ with $B$ does not contribute any new element that was not already in $B$. That is, $A \cup B = B$.