An Introduction to Proof through Analysis

This book discusses the construction and the writing of mathematical proofs. The authors tell a mathematical story as they cover the basic techniques of proof writing; the development of the number systems. The authors' first pass at developing the real numbers is slow and systematic. The second part covering logic, set theory, and the introduction of mathematical language is more careful and abstract. The third and final part repeats the development of the real numbers and moves on to topology and continuity. This book proceeds from the natural numbers to the rational numbers in a familiar way, and justifies the need for a rigorous definition of the Real Numbers. The authors aim the story to end with a mathematical climax; the Intermediate Value Theorem. This theorem, arguably, justifies the notion that the real numbers suffice to solve all geometric problems. This book examines how the algebraic properties of the rational numbers do not produce enough numbers to measure quantities, and also how a completeness axiom seems to produce all the numbers we need and nothing more. This book will be of interest to students in proof based undergraduate courses, such as an introduction to Real Analysis course.