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

Special Kinds of Theorems 91
which has all integer coefficients with an^O and ao 7^ 0, where n > 1.
Let z = p/q be written in its lowest terms, where q^O,
Then q divides a„ and p divides ao.
Proof: By hypothesis P{z) = 0. So,
Therefore,
a„p" + Un-ip'^'^q + • • • + aipq""-^ + ao^" = 0.
Thus, we can solve for a„p" and we obtain:
flnP" = -qian-ip""^ + • • • + aipg"-^ + ao^""^)
which can be rewritten as a„p" = ~^f, where t = a„_ip""^ H f-
a\pq^~^ + ao^"~^ is an integer.
This implies that q divides a„p". As p and ^ have no common factors,
q divides a„.
We can solve equation (*) for a^q^ to obtain:
which can be rewritten as a^q^ = -ps, where s = a„p"~^Han-\p^~^q
H h a\q^~^ is an integer. Thus, p divides ao^"-
Because p and ^ have no common factors, p must divide ao. •
LIMITS
The concepts of limits of functions and sequences are not easy ones to
grasp, and their definitions have been the results of the mathematical and
philosophical work of a number of mathematicians. This section includes
only the most basic ideas.
The formal definition of the limit of a function at a real number c is
usually stated as follows.
Definition. The real number L is said to be the limit of the function f{x)
at the point c, written as lim/(x) = L, if for every e > 0 there exists a
5>0 such that ii\x-c\<6 and x 7^ c, then |/(x) -L\<E.
This statement is very modern (less than 200 years old) in its precise
structure. For a long time mathematicians talked about vanishing