The principle of mathematical induction is a proof technique in which a statement, $~$P(n)$~$, is proven about a set of natural numbers $~$n$~$\. It may be best understood as treating the statements like dominoes: a statement $~$P(n)$~$ is true if the $~$n$~$\-th domino is knocked down\. We must knock down a first domino, or prove that a base case $~$P(m)$~$ is true\. Next we must make sure the dominoes are close enough together to fall, or that the inductive step holds; in other words, we prove that if $~$k \\geq m$~$ and $~$P(k)$~$ is true, $~$P(k+1)$~$ is true\. Then since $~$P(m)$~$ is true, $~$P(m+1)$~$ is true; and since $~$P(m+1)$~$ is true, $~$P(m+2)$~$ is true, and so on\.

I really like this domino analogy.

Also, I'd expect to see the word "all" somewhere in this first paragraph -- I think it's worth emphasizing the point that if we have the base case and the inductive step then the statement will be true for *all* of the numbers after the base case, just like all of the dominoes after the first one would fall down. I think the current final sentence of the intro paragraph doesn't make this clear enough.