310 PROPERTIES OF NUMBER SYSTEMS Chapter 9(6) *(i) Prove that if a, b E F with a^2 0 and b^2 0, then a < b if and only if
a2 < b2.
(ii) Prove that if a, b E F with a 2 0 and b 2 0, then a I b if and only if
a2 I b2.- (a) Let F be an ordered field with x E F. Prove:
(6) Suppose F is an ordered field with x, a E F.
(i) Prove that if a 2 0, then 1x1 I a if and only if -a I x I a.
(ii) Prove that if a^2 0, x I a, and -x I a, then 1x1 I a.- (a) Suppose F is an ordered field with x, y E F:
(i) Prove that 11x1 - lyll I IX - ~1 [Hint: Use (d) of Theorem^5 and (ii) of part
(b) of Exercise 5.1
(ii) Prove that if y # 0, then Illy1 = l/lyl.
*(iii) Prove that if y # 0, then Ix/yl = Ixl/lyl.
(iv) Prove that (x - yl I 1x1 + ly(.
(6) Suppose F is an ordered field, n E N, and x,, x,,... , x, E F. Prove that
1x1 + x, + - - - + x.1 I lxll + lx21 + - -. + lx.1. - (a) Prove Theorem 6.
(6) Prove that if F is an ordered field with x, y, z E F, then d(x, y) = d(x - z, y - z). - (a) Prove that if F is an ordered field, if x, y E F with x < y, then a(x + y) E F
and x < 3(x + y) < y.
(6) Conclude from (a) that an ordered field must contain an infinite number of
elements.
9.3 Completeness in an Ordered Field
Thus far, we have isolated R from N, Z, and C by means of abstract prop-
erties. It remains only for us to find a property by which the ordered
fields R and Q can be distinguished from each other. Now any ordered
field is infinite and has the property that between any two distinct elements
there is a third element (recall Exercise 8, Article 9.2); indeed, it is a familiar
fact that R and Q both have these properties. Thus rational numbers, like
real numbers, occur arbitrarily close to one another. Yet, the key to the
difference between the two fields is discovered by looking at them on a
microscopic level. In particular, a difference between R and Q begins to
surface when we consider an element of R - Q such as n. The infinite
sequence 3.1, 3.14, 3.14 1, 3.14 1 5,3.14159,... , consisting of successive deci-
mal approximations to n, has two properties of immediate interest: (1) Each
member is strictly less than n. (2) For any positive integer n, the nth number
in this sequence differs from n by less than lo-", so that any real number