Principle of Finite Induction/Proof 1
Theorem
Let $n_0 \in \Z$ be given.
Let $\Z_{\ge n_0}$ denote the set:
- $\Z_{\ge n_0} = \set {n \in \Z: n \ge n_0}$
Let $S \subseteq \Z_{\ge n_0}$ be a subset of $\Z_{\ge n_0}$.
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:
- $S = \Z_{\ge n_0}$
Proof
Let $\Z_{\ge n_0} := \set {n \in \Z: n \ge n_0}$.
Aiming for a contradiction, suppose $S \ne \Z_{\ge n_0}$.
Let $S' = \Z_{\ge n_0} \setminus S$.
Because $S \ne \Z_{\ge n_0}$ and $S \subseteq \Z_{\ge n_0}$, we have that $S' \ne \O$.
By definition, $\Z_{\ge n_0}$ is bounded below by $n_0$.
From Set of Integers Bounded Below by Integer has Smallest Element, $S'$ has a minimal element.
Let $k$ be this minimal element of $S'$.
By $(1)$ we have that:
- $n_0 \in S$
and so:
- $n_0 \notin S'$
Hence:
- $k \ne n_0$
and so:
- $k > n_0$
It follows that:
- $k - 1 \le n_0$
Because $k$ is the minimal element of $S'$:
- $k - 1 \notin S'$
and so:
- $k - 1 \in S$
But by $(2)$:
- $\paren {k - 1} + 1 = k \in S$
So we have:
- $k \in S$
and:
- $k \notin S$
Hence by Proof by Contradiction $S = \Z_{\ge n_0}$.
$\blacksquare$
Sources
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.6$: Mathematical Induction: Theorem $\text{A}.6$