If people asked you, what is the most elementary inequality you know? I bet your answer would be AM-GM. But in this series of posts I will try to show you the power that Cauchy Schwarz has over that of AM-GM. But as an introduction, let us first state and prove the theorem.
Cauchy Schwarz Inequality: Let and be two sequences of real numbers, then we have:
In particular, equality holds iff there exists for which for .
Proof: We will present proofs, one originating from analysis on the equality case, the other by wishful thinking on small cases of .
(i) Consider defining the following function :
We will expand this to get:
From our first way of representing , we can conclude that or
Equality holds if the equation has one root.■
(ii) Just remark that:
Let us note the following positive things regarding Cauchy-Schwarz:
it is effective in proving symmetric inequalities
Try to form squares
Helps to clear up square roots
Look forward to the next few posts to see applications of this extremely elegant inequality!