Binomial Theorem Let a and b be real numbers, and n a natural number. Recursive Definition Let S be a set. Then this construction determines a unique function h from N to S . Below are two recursive definitions, only one of which is valid. When first encountering proofs by induction, it seems that anything can be proved.
All birds are of the same color.
Assume that n birds are of the same color. Put one aside. There are n birds left, which, by assumption, are of the same color. For simplicity, say they are all black. Put the one bird back into the group, and take out another one.