20 Chapter 1 11 The Real Number System
(b) Three positive elements a, b, c of an ordered field form a geometric
b c
progression if their successive quotients are equal: a = b. Prove
that b = y'ac, if this square root exists.^4 [b is called the geometric
mean of a and c.](c) Three positive elements a, b, c of an ordered field form a harmonic
1
progression if their reciprocals form an arithmetic progression: -,;-
1 1 1 2ac- = - - -. Prove that b = --. [b is called the harmonic mean
a c b a+c
of a and c.]
(d) Three positive elements a, b, c of an ordered field form a quadratic
progression if their squares form an arithmetic progression: b^2 -a^2 = c^2 - b^2. Prove that b = ~' if this square root exists^4.
[b is called the quadratic mean of a and c.](e) P rove that in any ordered field, if 0 < a :::; b, and if the indicated
2ab CL a + b V a
2
+ b
2
square roots exist,^4 then a:::; a+ b :::; vab:::; -
2- :::;
2
:::; b.
1.3 Natural Numbers
In our development of ordered fields, we have not seen any familiar num-
bers except 0 and l. As we shall now see, all ordered fields must cont ain
lots of familiar numbers. First, they must all contain the "natural numbers,''
1, 2, 3, · · · , n, n + 1, · · ·. Before we can prove that, we must define the natural
numbers in a rigorous way.
Definition 1.3. 1 An inductive subset of an ordered field Fis a subset A ~
F with the properties:
(i) 1 EA, and
(ii) Vx E F, x E A:::::> x + 1 E A.Note that any ordered field F contains at least two inductive sets, for both
P and Fare inductive subsets of F. We shall see that there are many more.
- Square roots of positive elements do not necessarily exist in ordered fields. In the real
number system, however, they a lways exist (See Theorems 1.6.12, 2.5.11, 5.3.13, and Exercise
1.6-B.6).