ABSTRACT
The third chapter talks about inequalities in mathematics. It starts from defining several basic properties of inequalities and proceeds to prove many other familiar properties. This establishes the rules for manipulating inequalities, such as how they can be added or multiplied. Using this foundation, the chapter presents proofs of various inequalities, some of which are referred to in subsequent chapters. In particular, the third chapter introduces the concepts of arithmetic, geometric, harmonic, and quadratic means and proves the relationship between them. The chapter concludes with a proof of the Bernoulli inequality and the famous Cauchy-Schwarz inequality. That last proof is particularly interesting because the desired result pops up seemingly from nowhere!
