Principle of Mathematical Induction

Theorem
Let $\map P n$ be a propositional function depending on $n \in \Z$.

Let $n_0 \in \Z$ be given.

Suppose that:


 * $(1): \quad \map P {n_0}$ is true


 * $(2): \quad \forall k \in \Z: k \ge n_0 : \map P k \implies \map P {k + 1}$

Then:


 * $\map P n$ is true for all $n \in \Z$ such that $n \ge n_0$.

The principle of mathematical induction is usually stated and demonstrated for $n_0$ being either $0$ or $1$.

This is often dependent upon whether the analysis of the fundamentals of mathematical logic are zero-based or one-based.

Contexts
The Principle of Mathematical Induction can be introduced in a formal development of abstract algebra or mathematical logic in various contexts, and proved from first principles in each.

Also defined as
This principle can often be found stated more informally, inasmuch as the propositional function $P$ is referred to as "a statement about integers".

Also see

 * Principle of Finite Induction


 * Second Principle of Finite Induction
 * Second Principle of Mathematical Induction


 * Equivalence of Well-Ordering Principle and Induction