In order to reach conclusions and prove theorems, mathematicians rely upon a form of analysis that allows approximations for integrals, infinite sums, and solutions of differential equations; these approximations are expressed in terms of inequalities. This text addresses a difficult subject--one that is usually treated only in works for advanced students--at a level appropriate for undergraduates and even high school students, offering a background on inequalities that will prepare students to cope with the concepts of continuity, derivative, and integral. Unabridged republication of the edition published by Holt, Rinehart and Winston, New York, 1961.