Principle of Mathematical Induction

Theorem
Let $P \left({n}\right)$ be a propositional function depending on $n \in \N$.

Let $n_0 \in \N$ be given. ($n_0$ is often, but not always, zero or one.)

Suppose that:


 * $(1): \quad P \left({n_0}\right)$ is true
 * $(2): \quad \forall k \in \N: k \ge n_0 : P \left({k}\right) \implies P \left({k+1}\right)$

Then:


 * $P \left({n}\right)$ is true for all $n \ge n_0$.

This process is called proof by (mathematical) induction.

Proof
Consider $\N$ defined as a Peano structure.

The result follows from Principle of Mathematical Induction for Peano Structure: Predicate.