Date: Wed, 26 Nov 1997 09:35:56 -0500

From: Alex Bogomolny

Dear Ronald:

Congratulations on your success in math competition. The easiest way to prove Cauchy's Inequality is this. Let x be just a variable. Write

sum of (x*a_{i} + b_{i})^{2} =

and open parentheses. The result is a quadratic polynomial in x. Note that, by definition, this polynomial is never negative. This implies that its discriminant (the expression under the square root in the formula for its two roots) is non-negative. This is exactly the Cauchy's inequality.

Best regards,

Alexander Bogomolny