If is an integer, and is some property of integers (that is, for every integer , is a statement - meaning something that is either true or false - for instance, it could be the equation mentioned above). Then, if

**``base case'':**- is true and
**``induction step'':***whenever*is true, then is also true.

then is true for all integers .

The antecedent of the induction step, (the temporary assumption that
is true), is also called the *induction hypothesis*.

I like to think of induction as sort of like climbing a ladder. The base case is the bottom rung of the ladder and the induction step shows you how to climb from one rung to the next. If you know how to get to the bottom rung, and you know how to climb from one rung to the next, you can climb to every rung above the bottom one! Some people prefer a domino metaphor.

Zvezdelina Stankova-Frenkel 2001-11-18