A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$.
However, such matrices can also be characterized by the positivity of the principal minors.
A statement and proof can, for example, be found on wikipedia: http://en.wikipedia.org/wiki/Sylvester%27s_criterion
However, the proof, as in most books I have seen, is very long and involved. This makes sense in a book where you wanted to prove the other theorems anyway. But there has to be a much better way to prove it.
What is the "proof from the book" that positive definite matrices are characterized by their $n$ positive principal minors?