Principle of Finite Induction

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

Let $n_0 \in \N$ be given. ($n_0$ is often, but not always, zero or one.)

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 $S = \N$.

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

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