Proof by Induction
A proof technique for statements about natural numbers, using a base case and inductive step.
Principle
To prove \(\forall n \in \mathbb{N}: P(n)\):
- Base case: Prove \(P(0)\) (or \(P(1)\))
- Inductive step: Assume \(P(k)\) (inductive hypothesis), prove \(P(k+1)\)
Why It Works
The well-ordering principle of natural numbers guarantees that if the base case holds and each step propagates the property, then all natural numbers have the property.
Example
To be written.