epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Special Kinds of Theorems 59
B. There exists a rational number c such that a < c <b.
Proof: If a and b are two rational numbers, we can write a = ^ and
b = ^, where m, n, p, and q are integer numbers; n 7^ 0 and q^O.
As the average of two numbers is always between the two of them (see
Example 3 in the section on Equivalence Theorems), we can construct
the average of a and b and check whether it is a rational number.
Thus, we have:
a-{-b \ (m p\ mq-\-np
c = —— ' . I —
2\n qj Inq
The number mq + np is an integer because m, n, p, and q are integer
numbers; the number Inq is a nonzero integer because n and q are
integer numbers; n/0; and g/0. Therefore, the number c is a
rational number.
For the proof of the fact that a < c < fa, see the section on Equivalence
Theorems (Example 3). •
EXAMPLE 2. Let x be an irrational number. Then there is at least one
digit that appears infinitely many times in the decimal expansion of x.
Discussion: Because we do not know what the number x is we cannot even
hope to find explicitly which one of the possible ten digits is repeated
infinitely many times. Thus, the proof will not be a constructive one.
Proof: Using the contrapositive of the original statement, let us assume
that none of the decimal digits repeats infinitely many times. So, we can
assume that each digit k, with 0 < /c < 9, repeats nk times. Then the decimal
expansion of x has iV = no + ni H h ng + n9 digits. Therefore, x is a
rational number. Because the contrapositive of the original statement is true,
the statement itself is true. •
In Example 1, we were able to construct exphcitly the mathematical
object that satisfies the given requirements. In Example 2, we can only
estabUsh that an object exists, but we cannot explicitly give the procedure for
finding it.
EXAMPLE 3. A line passes through the points with coordinates (0,2)
and (2,6).
Discussion: We will reformulate the statement as: If (0,2) and (2,6) are two
points in the plane, then there is a Une passing through them. We will prove
this statement in two ways: by finding the equation of the Une explicitly, and
by using a theoretical argument.