Principle of Mathematical Induction/One-Based/Proof 3
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Theorem
Let $\map P n$ be a propositional function depending on $n \in \N_{>0}$.
Suppose that:
- $(1): \quad \map P 1$ is true
- $(2): \quad \forall k \in \N_{>0}: k \ge 1 : \map P k \implies \map P {k + 1}$
Then:
- $\map P n$ is true for all $n \in \N_{>0}$.
Proof
We have that Natural Numbers are Non-Negative Integers.
Then we have that Integers form Well-Ordered Integral Domain.
The result follows from Induction on Well-Ordered Integral Domain.
$\blacksquare$
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $3$: The Integers: $\S 9$. The Principles of Induction: Theorem $14$