3 Ordinary (not so strong) induction

This is like strong induction, but you can only assume $P(n-1)$ in order to prove $P(n)$ in the induction step. In other words,

$(\forall i \in I: (a \le i \le b) \rightarrow P(i)), (\forall k \in I, (b < k) ... ... (P(k-1) \rightarrow P(k)) \vdash (\forall n \in I: (n \ge a) \rightarrow P(n))$