Principle of Finite Induction

Theorem
Let $S \subseteq \N$ be a subset of the natural numbers.

Let $n_0 \in \N$ be given.

Suppose that:


 * $(1): \quad n_0 \in S$


 * $(2): \quad \forall n \ge n_0 : n \in S \implies n + 1 \in S$

Then:


 * $\forall n \ge n_0$: $n \in S$.

In particular, if $n_0 = 0$, then $S = \N$.

The principle of mathematical induction is usually stated and demonstrated for $n_0$ being either $0$ or $1$.

This is often dependent upon whether the analysis of the fundamentals of mathematical logic are zero-based or one-based.

Proof
Consider $\N$ defined as a naturally ordered semigroup.

The result follows directly from Principle of Mathematical Induction for Naturally Ordered Semigroup: General Result.