The Cauchy–Schwarz Master Class: Starting with Cauchy

TL;DR: In this paper, the typical chapter is built around the solution of a small set of challenge problems, usually drawn from one of the world's famous mathematical competitions, but more often a problem is chosen because it illustrates a mathematical technique of wide applicability.

Abstract: Cauchy's inequality for real numbers tells us that a 1 b 1 + a 2 b 2 + · · · + a n b n ≤ a 2 1 + a 2 2 + · · · + a 2 n b 2 1 + b 2 2 + · · · + b 2 n , and there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics. A central aim of this course — or master class — is to suggest a path to mastery of this inequality, its many extensions, and its many applications — from the most basic to the most sublime. The Typical Plan The typical chapter in this course is built around the solution of a small set of challenge problems. Sometimes a challenge problem is drawn from one of the world's famous mathematical competitions, but more often a problem is chosen because it illustrates a mathematical technique of wide applicability. Ironically, our first challenge problem is an exception. To be sure, the problem hopes to offer honest coaching in techniques of importance, but it is unusual in that it asks you to solve a problem that you are likely to have seen before. Nevertheless, the challenge is sincere; almost everyone finds some difficulty directing fresh thoughts toward a familiar problem. Problem 1.1 Prove Cauchy's inequality. Moreover, if you already know a proof of Cauchy's inequality, find another one! Coaching for a Place to Start How does one solve a problem in a fresh way? Obviously there cannot be any universal method, but there are some hints that almost always help. One of the best of these is to try to solve the problem by means of a specific principle or specific technique. Here, for example, one might insist on proving Cauchy's inequality 1