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Proof by Induction

A proof technique for statements about natural numbers, using a base case and inductive step.

Principle

To prove \(\forall n \in \mathbb{N}: P(n)\):

  1. Base case: Prove \(P(0)\) (or \(P(1)\))
  2. 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.

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