User:Prime.mover/Proof Structures/Proof by Complete Finite Induction

Proof by Complete Finite Induction
The proof will proceed by the Principle of Complete Finite Induction on $\Z_{>0}$.

Let $S$ be the set defined as:
 * $S := \set {n \in \Z_{>0}: ...}$

That is, $S$ is to be the set of all $n$ such that:

Basis for the Induction
We have that:

(proof that $1 \in S$)

So $1 \in S$.

This is the basis for the induction.

Induction Hypothesis
It is to be shown that if $j \in S$ for all $j$ such that $0 \le j \le k$, then it follows that $k + 1 \in S$.

This is the induction hypothesis:
 * $\forall j \in \Z_{>0}: 1 \le j \le k: \text {expression for $j$}$

It is to be demonstrated that it follows that:
 * $\text {expression for $k + 1$}$

Induction Step
This is the induction step:

So $\forall j \in S: 0 \le j \le k: j \in S \implies k + 1 \in S$ and the result follows by the Principle of Complete Finite Induction:


 * $\forall n \in \Z_{>0}: ...$