Principle of Finite Induction/One-Based/Proof 1
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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 $1 \text{-} 2$