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 $A = \{n \in \N| P(n)\}$. We show that $A$ is an inductive set.

By $(1)$: $i \in A$, $\forall 1 \leq i \leq d$.

Assume $\{1,2,\ldots,x\} \subset A$, $x \geq d$. Then $\forall k\in \N,x-(d-1)\leq k < x + 1:P(k)$ by definition of $A$, thus $P(x+1) \Rightarrow x+1\in A$. Thus $A$ is an inductive set.

Thus by the fifth axiom of Peano, $A = \N \Rightarrow P(n)$ $\forall n\in \N$.