Principle of Mathematical Induction

Axiom 1:  Principle of Mathematical Induction

A proof by induction consists of 2 steps:

(1)  Base Case:  Proves that the statement is true for one case independent of all other cases.

(2)  Inductive Case: Proves that if the statement holds for any case $n=k$, then it must also hold for $n=k+1$

The principle can be understood with the analogy of a ladder.  if we can climb on the bottom rung of a ladder and for each rung, we can climb to the next, then it follows that we can climb as high as we like up to the top of the ladder. 

This principle is very important in the establishment of advanced mathematical arguments.