The formulas on the previous pages, for the th term and for the sum of the first terms, are claims about every positive integer . Mathematical induction is a method for proving claims of that kind.
We know there is no largest integer, because when we add to an integer we get .
Here is a family of assertions , one for each positive integer , with the first three written out:
The principle of mathematical induction
Induction:
Prove for .
Show that if it is true for any , then it is true for .
Thus it is true for all positive integers.
Theorem 11.4.1 (Principle of mathematical induction). Let be an assertion involving an integer . We conclude that is true for every if we can:
(a) prove is true;
(b) let be an arbitrary but fixed integer , assume that is true, and prove that is true.