10.3 OUTLINE OF THE CONSTRUCTION OF THE REALS 361
You need to use both the fact that {a,) is Cauchy and the fact that {a,) is
eventually bounded away from zero.)
- Prove that the mapping q -+ [q] of Q into R, defined in the paragraph follow-
ing the proof of Theorem 5, is one to one, not onto, and preserves addition and
multiplication in Q. Prove also that the image of this mapping, a proper subset of
R, is closed under addition and multiplication.
- Prove that the definition of positivity in R, given in Definition 6, is well defined
(see the precise formulation of well-definedness in this context following Definition
6).
- Prove that any real number x lies between two consecutive integers; that is, prove
that, given x E R, there corresponds an integer n such that n - 1 < x < n. (Hints:
First, use the Archimedean property of R to prove that x must necessarily lie
between two integers, say, m and p. Having accomplished this, use the well-ordering
principle for N to show that x lies between two consecutive integers in the set
{m,m + 1,... , m + (p - m)).)
