Mathematical induction is a method of proof by which a statement about a variable can be demonstrated to be true for all integer values of that variable greater than or equal to a specified integer (usually 0 or 1).
An example of such a statement is:
- The number of possible pairings of n distinct objects is (for any positive integer n).
A proof by induction proceeds as follows:
- The statement is proved for the first possible value of n (usually 0 or 1, but other "starting values" are possible).
- The statement is assumed to be true for some fixed, but unspecified, value n and this assumption is used to prove that the statement is true for (the latter statement is simply the original statement with n replaced by ).
- The statement is then concluded to be true for all relevant values of n (all nonnegative values or all positive values, depending).
That the conclusion in step 3 above follows from steps 1 and 2 is the principle of mathematical induction.
More formally, given a proposition about the integer-valued variable n that is to be proved for , the following must be proved.
The conclusion is then
- for .