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

Special Kinds of Theorems 49
3. Using the inductive hypothesis, prove that the statement is true for the
next number in the collection (deductive step).
Test that the
statement is true for
the smallest number
in the collection.
I
State the inductive
hypothesis for an arbitrary
number in
the collection.
' I '
Prove that the
statement is true
for the next number
in the collection.
At first, the construction of the proof by induction might seem quite
pecuhar. We start by checking that the given statement is true in a special
case. We know that we cannot stop here because examples are not proofs.
Then we seem to "trust" the statement to be true temporarily, and then we
check its strength by using deductive reasoning to see if the truth of the
result can be extended one step further. If the statement passes this last test,
then the proof is complete. This construction works Hke a row of dominoes;
when the first one is knocked down, it will knock down the one after it, and
so on until the entire row is down.
The three steps show that the statement is true for the first number and
that whenever the statement is true for a number it will be true for the next.
The fact that this extension process can be extended indefinitely from the
smallest number requires an in-depth explanation that is beyond the
purpose of this book. Indeed, the technique of mathematical induction is
founded on a very important theoretical result—namely, the principle of
mathematical induction, usually stated as follows:
Let P{n) represent a statement relative to a positive integer n. If:
1. P{t) is true, where t is the smallest integer for which the statement can
be made,
2. whenever P{n) is true, it follows that P(n + 1) is true as well,
then P(n) is true for all n>t.