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

Special Kinds of Theorems 43
Suppose that the angles ICAB and ICBA are equal. Consider the two
triangles ADC and CDB, obtained by constructing the segment CD, which
starts from the third vertex and is perpendicular to the base, AB.
The angles lADC and IBDC are equal, as they are right angles. The two
angles at the vertices A and B are equal by hypothesis; therefore, the angles
lACD and IDCB are equal as well. Thus, the two triangles ADC and CDB
are similar.
Moreover they have the side CD in common. Thus, the triangles are
congruent. In particular, the sides AC and CB are equal. •
EXAMPLE 5. Let / be a positive function defined for all real numbers.
Then the following statements are equivalent:
1. /is a decreasing function.
2. The function g, defined as g(x) = l//(x), is increasing.
3. The function h, defined as h{x) = -/(x), is increasing.
4. The function /c„, defined as kn(x) = nf{x\ is decreasing for all positive
real numbers n.
(See front material in the book for the definitions of increasing and
decreasing functions.)
Proof: We will prove that statement 1 implies statement 2, statement 2
implies statement 3, statement 3 impHes statement 4, and statement 4 implies
statement 1, as shown in the following diagram:
1 -> 2
t ;
4 ^ 3
Part 1. If 1, then 2.
As/is decreasing, given any two real numbers xi and X2 such that xi < X2, it
follows that/(xi)>/(x2). Therefore, l//(xi)<l//(x2). This means that
G{x\)<g{x2\ and g is an increasing function.
Part 2. If 2, then 3.
By definition of the functions used, h{x) = —\/g{x). As g is an increasing
function, for every two real numbers x\ and X2 such that xi < X2, it follows
that g{xi)<g{x2). Thus, \/g{xi)>\lg{x2\ and -\/g{xi)< -l/^fe). This
implies that h{x\)<h{x2\ so h is an increasing function.
Part 3. If 3, then 4.
By definition of the functions used, kn(x) = n(-h(x)) = -nh(x). As h is
an increasing function, for every two real numbers xi and X2 such that
xi < X2, it follows that h(xi)<h(x2). Therefore, -h(xi}>h(x2), and