question topic: Dedekind cut property from real analysis. BSc level. Boook name: A First Course in Real AnalysisBy Sterling K. Berberian

2. With notations as in Exercise 1, the pair (A, B) is called a (Dedekind) cut of the rational field Q. (i) Show that the formulas A={reQ: rs0]U {reQ: r> 0 and r? 0 and r> > 2} define a cut (A, B) of the rationals. {Hint: There is no rational number r with r? = 2.} (ii) Let y be the real number provided by Exercise 1. Show that y = {Hint: As shown in the proof of 2.2.4, there is no largest positive number whose square is 2.} 2. First Properties of R Dedekind's stroke of genius: Define v2 to be this cut (A, B) of the rationals. There are still many loose ends to be looked after (defining sums and products of cuts, regarding cuts (*) and (**) that lead to the same real number as being 'equal', etc.), but these are largely housekeeping chores. The decisive stroke is the first: something new ( v2 ) has been defined in terms of something old (a strategically chosen pair of subsets of Q)