An Introduction to Proof through Real Analysis

Book Preface

Why proof?

For most people, Mathematics is about using mathematical facts to solve practical problems. Users of Mathematics are rarely concerned about why the methods work and care only that they do work. To too many people, Mathematics is a collection of arcane techniques known only to a select few with “math brains.” It is troublesome when those arcane techniques that confuse people are differentiation, integration, or matrix manipulation. It is downright frightening when the confusing problems are adding fractions or computing a restaurant tip.The worst way to view Mathematics is as a long collection of hard-to-remember techniques for solving specific problems. A much better way is to think of Mathematics as an organization of basic ideas that can solve all sorts of problems as needed.When you understand what Mathematics actually means, you can use that understanding to produce your own problem solving techniques. The key to understanding any piece of Mathematics (or anything else for that matter) is to understand why it works the way it does.

Since the ancient Greeks first studied Mathematics in a careful way, the subject has been built on deductive proof. Mathematical results are accepted as facts only after they have been logically proved from a few basic facts. Once mathematical facts are established, they can be used to solve practical and theoretical mathematical problems. Mathematicians have two reasons for proving a mathematical statement rigorously: first, to be sure that the result is true, and second, to understand when and how it works.

Following the ancient Greek process, mathematicians want a proof for everything – whether it is on the cutting edge of mathematics and science or it is an apparently obvious fact about grade school arithmetic. The idea is to understand why a mathematical result is true and to move on to what you know because it is true. Most of the Mathematics we see in school is about the “moving on” variety. Once school children understand the connection between combining small groups of objects and adding numbers, they can move on to the arithmetic algorithm of adding larger numbers.Thus, is just the theoretical way to combining 278 objects and 394 objects and counting the combination. Once school children understand the connection between groups of groups and multiplication, they can learn the algorithm for multiplication.Then is just the theoretical way of counting 35 rows of 257 objects.

At the very beginning, every child is given some simple justifications for the validity of these algorithms. The strong belief among math educators and education researchers is that students who understand those justifications best are the students that will learn the algorithms best. Granted in the long run, it is a child’s ability with the algorithm that is considered most important. In time, greater facility with the algorithms supplants a person’s need for the logic behind those algorithms. But the complete understanding of the operation behind the algorithm is always essential for its proper use in odd situations.