Examination of the above proof shows that

if and only if there is a fixed number x such that a_ix + b_i = 0 for all i, that is, the a_i and b_i are proportional.

The hypothesis for the inequality on the arithmetic and geometric means is that the numbers are all positive. The numbers in the Chauchy-Schwarz Inequality need not be positive. In fact,

is unaltered by changes in the signs of the a_i and b_i, while

is largest when all the signs are positive.
 
 

Monday, June 22, 1998