Induction/Variable/Relevance/Remark

Proofs by induction occur again and again. The condition, that this method of proof can be applied, is that we have a scheme of statements, which depend on the (variable) natural number ${}n$. This natural number ${}n$ is called the induction variable, we do induction over the induction variable ${}n$. In the statement itself, arbitrary mathematical objects may occur and the natural number ${}n$ can have many different meanings. It can be the exponent of a real number (see fact or fact), the degree of a polynomial (as in fact), the degree of differentiability (see exercise) or the number of vectors (as in fact).