Let and . Prove that is a Dedekind cut and also that it has the property ; that is, the square of is . Note: This seems to be surprisingly tricky, as pointed out by Linda Hill and Robert J. Fisher at Idaho State University. Their solution is available from them or from the author.
Obviously a homework problem.
The only real difficulty is in showing that the set does NOT have a largest member.
Suppose x is the largest member of the set of all rational numbers whose square is less than 2. Since you can assume 1.4< x. Since you can assume x< 1.5. Let (d is, of course, rational). Then . Now find an integer, n such that showing that x is NOT the largest number in that set.
take a peek at wikipedia, eh?!