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

Collection of Proofs 119
where k is an integer and k>0. Then,
i±V8feT3
s = 2 •
Because s is positive, we will only consider:
l + ^/8feT5
' = 2 •
Clearly, 8/c + 5 is never a perfect square (it is equal to 5, 13, 21, 37,...).
Therefore, s is not an integer, and the given statement is true. •
THEOREM 6. All math books have the same number of pages.
Alleged Proof. We will prove by induction on n that all sets of n math
books have the same number of pages.
Step 1: Let n= 1. If X is a set of one math book, then all math books
in X have the same number of pages.
Step 2: Assume that in every set of n math books all the books have
the same number of pages.
Step 3: Now suppose that X is a set of n + 1 math books. To show that
all books in X have the same number of pages, it suffices to show that, if a
and b are any two books in X, then a has the same number of pages as b.
Let ybe the collection of all books in X, except for a. Let Z be the collection
of all books in X, except for b. Then both Y and Z are collections of n
books. By the inductive hypothesis, all books in 7have the same number of
pages, and all books in Z have the same number of pages. So, if c is a book in
both 7 and Z, it will have the same number of pages as a and b. Therefore,
a has the same number of pages as b. •
THEOREM 7. Ifn>0
and a is a fixed nonzero real number, then a^=l.
Alleged Proof. By mathematical induction.
Step 1: Let n = 0. Then, by definition, a^=l.
Step 2: Assume that a'^ = 1 for all 0 < /c < n (strong inductive hypothesis).
Step 3: Let us work on n + 1. By the rules of algebra:
fl"-i 1
Therefore, the conclusion is proved.
•