6.3 EXISTENCE AND UNIQUENESS (OPTIONAL) 213You may recall the notation (3!x)(p(x)) and the accompanying formal
definition of "unique existence" from Exercise 9, Article 3.3. Specific in-
stances of the themes of existence and uniqueness occur in a variety of
mathematical settings, as the following example shows.
EXAMPLE (^1) (a) Show that there is a unique real solution to the equation
x - 5 = (existence uniqueness; see Example 2).
(b) Show that there exists an "upper bound" for the interval [0, 11
(existence only; see Example 7).
(c) Show that iff is a function defined on an open interval con-
taining a point a, then lim,,, f (x), if it exists, is unique (uniqueness only;
see Example 5).
(d) Show that if m and n are integers, not both zero, then they have
a unique greatest common divisor (existence and uniqueness; see Example
10).
(e) Show that there exists an infinite number of primes (existence
only; see Example 9).
Techniques of proof pertaining to these two themes are different; yet we
treat them in the same article because, as parts (a) and (d) of Example 1
illustrate, the two themes often occur together as two parts of a single
problem, one part being the "flip side" of the other. Indeed, the dual theme
of existence and uniqueness runs through mathematics at all levels, begin-
ning in high school algebra. Another reason for treating them together
is that sometimes the process by which uniqueness is proved provides the
key to the proof of existence, as in the following example.
EXAMPLE 2 Show that both the equations 7x - 5 = 0 and x - 5 =
have unique real solutions.
Solution Both these equations may be approached through normal al-
gebraic manipulations, directed toward solving the equation, namely:
7x - 5 = 0 and x-5 = ,/o
- 7x = 5 * (x - 5)2 = X + 7
- x=+ * x2-lOx+25=x+7
=> x2 - 11x + 18 =O - (x - 9)(x - 2) = 0
At first glance, the second equation appears to have two solutions, but
substitution of both candidates into the original equation reveals that
x = 2 does not satisfy the equation, since - 3 # &. This example illus-
trates the general fact that the process of solving an equation algebra-
ically never proves that any number actually solves the equation, and so,