User:Prime.mover/Proof Structures/Proof by Complete Induction
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Proof by Complete Induction
The proof proceeds by strong induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
- $proposition_n$
$\map P 0$ is the case:
- $proposition_0$
Thus $\map P 0$ is seen to hold.
Basis for the Induction
$\map P 1$ is the case:
- $proposition_1$
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that if $\map P j$ is true, for all $j$ such that $0 \le j \le k$, then it logically follows that $\map P {k + 1}$ is true.
This is the induction hypothesis:
- $proposition_k$
from which it is to be shown that:
- $proposition_{k + 1}$
Induction Step
This is the induction step:
\(\ds \) | \(=\) | \(\ds \) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(=\) | \(\ds \) |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Second Principle of Mathematical Induction.
Therefore:
- $proposition_n$