Definition:Mathematical Induction

Proof Technique
Mathematical induction is a proof technique which works in two steps as follows:


 * $(1): \quad$ A statement $Q$ is established as being true for some distinguished element $w_0$ of a well-ordered set $W$.


 * $(2): \quad$ A proof is generated demonstrating that if $Q$ is true for an arbitrary element $w_p$ of $W$, then it is also true for its immediate successor $w_{p^+}$.

The conclusion is drawn that $Q$ is true for all elements of $W$ which are successors of $w_0$.

It is established in a number of contexts, according to how it is to be used.

Also see

 * Do not confuse with Definition:Inductive Argument.