Bridge to Abstract Mathematics: Mathematical Proof and Structures

2l6 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6

a point xo in the domain off that is within both 6, and 6, distance of

a. For this xo, we may note that

IL, - L2I = IL, - f@O) + Do) - L2I

5 ILl - f(x0)l + If(x0) - L2I

= If(x0) - ~ll + If(x0) - L2I

< ~/2 + P/2

= P

We have established that, for an arbitrary p > 0, IL, - L,I i p, so that

L, = L2, as desired. 0

Among the exercises [Exercise 6(b)], you will be asked to mimic the
preceding proof to show that an infinite sequence has at most one limit.
Note also that the result of Example 5 can be derived through an indirect
argument that is quite worthwhile. In Exercise 6(a) you are asked to provide
such a proof.
In the next example we illustrate the type of uniqueness proof in which
we have to prove that a specific named object a is the only object satisfying
a given property p(x).

EXAMPLE 6 Show that the empty set (25 is the only set W that may have
the property that there exist distinct sets A and B such that A x W =
B x W.

Solution Suppose that W is an arbitrary set such that A x W = B x W

for some pair of distinct sets A and B. We claim W = (25. Proceeding

indirectly, we note that if W # 0 and A x W = B x W, then by Example

9, Article 5.2, we may conclude A = B, contradicting our assumption.

Thus W = (25, as claimed.

Note that to prove the theorem "(25 is the only set W having the prop-
erty... ," we must verify additionally that there do indeed exist distinct
sets A and B such that A x (25 = B x (25. (This is the existence side of
the existence-uniqueness duality.) Of course, this is true since A x 0 =
B x 0 = 0 for any sets A and B, so that we may, for example, let A =
(1) and B = (2).
Proofs of uniqueness similar in type to the one in Example 6 are found
in Exercises 3(a, b), 4, and qc), among others.

EXISTENCE

There are essentially two elementary approaches to proving existence of an

object satisfying a given property or collection of properties. One is the

direct approach; we prove existence by producing an object of the desired

type. In many cases we produce such an object by naming one explicitly,

as in Examples 7 and 8. Existence proofs likely to be encountered by under-

graduates in which this can be done are often of the easier variety. In more