Third Principle of Mathematical Induction

Theorem
Let $P \left({n}\right)$ be a propositional function depending on $n \in \N$.

If:


 * $(1): \quad P \left({n}\right)$ is true for all $n \le d$ for some $d \in \N$
 * $(2): \quad \forall m \in \N: \left({\forall k \in \N, m \le k < m + d: P \left({k}\right)}\right) \implies P \left({m + d}\right)$

then $P \left({n}\right)$ is true for all $n \in \N$.

Proof
Let $A = \left\{ {n \in \N: P \left({n}\right)}\right\}$.

We show that $A$ is an inductive set.

By $(1)$:
 * $\forall 1 \le i \le d: i \in A$

Let:
 * $\forall x \ge d: \left\{ {1, 2, \dotsc, x}\right\} \subset A$

Then by definition of $A$:
 * $\forall k \in \N: x - \left({d - 1}\right) \le k < x + 1: P \left({k}\right)$

Thus $P \left({x + 1}\right) \implies x + 1 \in A$

Thus $A$ is an inductive set.

Thus by the fifth axiom of Peano:
 * $\forall n \in \N: A = \N \implies P \left({n}\right)$