18 Chapter 1 • The Real Number System
Theorem 1.2.15 (Triangle Inequalities) For all x, y in an ordered field
F,
(a) Ix+ YI ::::; lxl + IYI;
(b) lxl - IYI::::; Ix -yl;
(c) llxl - IYll ::::; Ix - YI;
(d) llxl - IYll::::; Ix+ YI·
Proof. (a) First, recall that -lxl ::::; x ::::; lxl, and -IYI ::::; y ::::; IYI· Thus, by
Theorem 1.2.10 (b),
-lxl-IYI::::; x+y::::; lxl + IYI, or
- (lxl + IYI)::::; x + Y::::; lxl + IYI·
But lxl + IYI is a nonnegative element of F. Thus, by Theorem 1.2.14 (a),
Ix+ YI::::; lxl + IYI·
(b) By Part (a), lxl = l(x - y) +YI ::::; Ix - YI+ IYI· Subtracting IYI from
both sides, we have lxl - IYI ::::; Ix -YI·
( c) Exercise 3.
( d) Exercise 3. •
INTERVALSDefinition 1.2.16 (Intervals) Let F be an ordered field. We first define closed
intervals and then extend this definition to define arbitrary intervals.(a) 'Va, b E F, we define the closed interval [a, b] to be the set
[a,b] = {x E F: a::::; x::::; b}.
Note that we do not require that a < b in this definition. Thus, for example,
[a, a]= {a}, and
[2, l] = 0, since 2::::; x::::; 1 is impossible.
(b) In general, an interval in F is any subset I <;:;; F such that [a, b] <;:;; I
whenever a, b EI.
That is , an interval is a set that always contains the entire closed interval
between any two of its points. It thus would make sense to say that an interval
is a "convex set" in an ordered field. The following theorem specifies exactly
which sets are intervals.
Theorem 1.2.17 (Intervals) In an ordered field F, the following sets are
intervals:(a) [a,b] = {x E F: a::::; x::::; b}; (This could be {a} or 0.)
(b) (a, b) = { x E F : a < x < b} ; ( This could be 0 .)