Principle of Finite Induction/Proof 2

Theorem

Let $S \subseteq \Z$ be a subset of the integers.

Let $n_0 \in \Z$ 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:

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

That is:

$S = \set {n \in \Z: n \ge n_0}$

Proof

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

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

$\blacksquare$