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.

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".