Date: Wed, 26 Nov 1997 20:54:51 +0800

From: Calvin Lui

Dear Alexander,

Hi,I am Ronald. I sorry that I didn't write back to you. I was very busy to prepare for a mathematics competition in my school and I am happy to tell you that I got the 4th. Last time,you asked me how much do I know about integration. Right? I learned the basic concept of integration. But I am happy to tell you that I finally found that how to prove the volume of a pyramid.

However, I want to ask a very that question: i want to know how to prove the Cauchy Inequality. Let a_{1}, a_{2}, ..., a_{n} and b_{1}, b_{2}, ..., b_{n} be real numbers. Show that:

(a_{1}b_{1} + ... + a_{n}·b_{n})^{2} ≤
(a_{1}^{2} + ... + a_{n}^{2}) · (b_{1}^{2} + ... + b_{n}^{2})

I have tried to expand it. But I found out nothing. Actually I have read a lot of theorem about inequality. But I still can't find the answer. Could you help me?

Thank You very much! I am looking forward to hearing from you!

Best,

Ronald