Principle of Finite Induction/One-Based/Proof 1

Theorem

Let $S \subseteq \N_{>0}$ be a subset of the $1$-based natural numbers.


Suppose that:

$(1): \quad 1 \in S$
$(2): \quad \forall n \in \N_{>0} : n \in S \implies n + 1 \in S$


Then:

$S = \N_{>0}$


Proof

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

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

$\blacksquare$


Sources

  • 1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Theorem $1 \text{-} 2$
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