Let us examine this more closely. Since this case is correct. Suppose we already know that , for an arbitrary, but fixed number . We have so that this case is also correct, under the above assumption. We did indeed give a correct proof, based on the mathematical induction.
Mathematical induction is a powerful proof technique closely related to recursion.
It is a principle because it can not be derived from other facts, but must be assumed as a basic property of the natural numbers. The induction principle can be generalized so as to apply to integers , rather than natural numbers. Basis step . It is part of the inductive step. Basis step .