You have asked a very complicated question. The complication results from the wide varieties of notation is use when discussing Dedekind Cuts, Your best bet is find a good text in your library that uses Cuts. A presently popular text is Elementry Analysis by Ken Ross it has a very brief discussion. A very full development can be found in John Randolph’s Basic Real and Abstract Analysis. It has some 15 pages on Cuts but it is an older text.