Principle of Finite Induction

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

The principle of finite 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.

Contexts
The Principle of Finite Induction can be introduced in a formal development of abstract algebra or mathematical logic in various contexts, and proved from first principles in each.

Also see

 * Principle of Mathematical Induction


 * Second Principle of Finite Induction
 * Second Principle of Mathematical Induction