In this article we study on various proofs and applications of the Cauchy inequality. Prof, Park H.S. claims that though “Mathematics for teachers” is not compulsory materials in school, it gives to mathematics teacher feeling of reassurance in the process of teaching mathematics. In our work, we review some materials concerned with Cauchy inequality, elaborate these, and develop materials for the mathematics teacher. We think that our materials are suitable to contents of “Mathematics for teachers”.