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Section 0.2 About analysis

Analysis is the branch of mathematics that deals with inequalities and limits. The present course deals with the most basic concepts in analysis. The goal of the course is to acquaint the reader with rigorous proofs in analysis and also to set a firm foundation for calculus of one variable (and several variables if volume II is also considered).

Calculus has prepared you, the student, for using mathematics without telling you why what you learned is true. To use, or teach, mathematics effectively, you cannot simply know what is true, you must know why it is true. This course shows you why calculus is true. It is here to give you a good understanding of the concept of a limit, the derivative, and the integral.

Let us use an analogy. An auto mechanic that has learned to change the oil, fix broken headlights, and charge the battery, will only be able to do those simple tasks. He will be unable to work independently to diagnose and fix problems. A high school teacher that does not understand the definition of the Riemann integral or the derivative may not be able to properly answer all the students' questions. To this day I remember several nonsensical statements I heard from my calculus teacher in high school, who simply did not understand the concept of the limit, though he could “do” the problems in the textbook.

We start with a discussion of the real number system, most importantly its completeness property, which is the basis for all that comes after. We then discuss the simplest form of a limit, the limit of a sequence. Afterwards, we study functions of one variable, continuity, and the derivative. Next, we define the Riemann integral and prove the fundamental theorem of calculus. We discuss sequences of functions and the interchange of limits. Finally, we give an introduction to metric spaces.

Let us give the most important difference between analysis and algebra. In algebra, we prove equalities directly; we prove that an object, a number perhaps, is equal to another object. In analysis, we usually prove inequalities, and we prove those inequalities by estimating. To illustrate the point, consider the following statement.

Let \(x\) be a real number. If \(x < \epsilon\) is true for all real numbers \(\epsilon > 0\text{,}\) then \(x \leq 0\).

This statement is the general idea of what we do in analysis. Suppose next we really wish to prove the equality \(x = 0\text{.}\) In analysis, we prove two inequalities: \(x \leq 0\) and \(x \geq 0\text{.}\) To prove the inequality \(x \leq 0\text{,}\) we prove \(x < \epsilon\) for all positive \(\epsilon\text{.}\) To prove the inequality \(x \geq 0\text{,}\) we prove \(x > -\epsilon\) for all positive \(\epsilon\text{.}\)

The term real analysis is a little bit of a misnomer. I prefer to use simply analysis. The other type of analysis, complex analysis, really builds up on the present material, rather than being distinct. Furthermore, a more advanced course on real analysis would talk about complex numbers often. I suspect the nomenclature is historical baggage.

Let us get on with the show...

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