The simplest and most common form of mathematical induction proves that a statement holds for all natural numbers n and consists of two steps:
To understand why the two steps are in fact sufficient, it is helpful to think of the domino effect: if you have a long row of dominos standing on end and you can be sure that
Example:
m(m + 1) 1 + 2 + ... + m = -------- 2
m(m + 1) 1 + 2 + ... + m + (m + 1) = -------- + (m + 1) 2
m(m + 1) 2(m + 1) (m + 2)(m + 1) = -------- + -------- = -------------- 2 2 2
((m + 1))((m + 1) + 1) 1 + 2 + ... + (m + 1) = ---------------------- 2
This type of proof can be generalized in several ways. For instance, if we want to prove a statement not for all natural numbers but only for all numbers greater than or equal to a certain number b then the following steps are sufficient:
Another generalization allows that in the second step we not only assume that the statement holds for n = m but also for all n smaller than or equal to m. This leads to the following two steps:
This can be used, for example, to show that fib(n) = [Φ^{n} - (-1/Φ)^{n} ] / 5^{1/2} where fib(n) is the n^{th} Fibonacci number and Φ = (1 + 5^{1/2}) / 2 (the socalled Golden mean). Since fib(m + 1) = fib(m) + fib(m - 1) it is straightforward to prove that the statement holds for m + 1 if we can assume that it already holds for both m and m - 1. Also for this generalization it holds that it is in fact just a special case of the first form; let P(n) be the statement that we intend to prove then proving it with these rules is equivalent with proving the statement ' P(m) for all m <= n ' for all natural numbers n with the first two steps.
The last two steps can be reformulated as one step:
This is in fact the most general form of mathematical induction and it can be shown that it is not only valid for statements about natural numbers, but for statements about elements of any well-founded set, that is, a set with a partial order that contains no infinite descending chains (where < is defined such that a < b iff a <= b and a ≠ b).
This form of induction, when applied to ordinals (which form a well-ordered and hence well-founded class), is called transfinite induction. It is an important proof technique in set theory, topology and other fields.
Proofs by transfinite induction typically need to distinguish three cases: