Principle of Finite Induction/Proof 2
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The validity of the material on this page is questionable.
This only takes on board a subset of $\N$, where we need a subset of $\Z$
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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
This only takes on board a subset of $\N$, where we need a subset of $\Z$
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Consider $\N$ defined as a naturally ordered semigroup.
The result follows directly from Principle of Mathematical Induction for Naturally Ordered Semigroup: General Result.
$\blacksquare$
