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## Über dieses Buch

This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
This text is an honours-level undergraduate introduction to real analysis: the analysis of the real numbers, sequences and series of real numbers,and real-valued functions.
Terence Tao

### Chapter 2. Starting at the beginning: The natural numbers

Abstract
In this text, we will review the material you have learnt in high school and in elementary calculus classes, but as rigorously as possible. To do so we will have to begin at the very basics - indeed, we will go back to the concept of numbers and what their properties are. Of course, you have dealt with numbers for over ten years and you know how to manipulate the rules of algebra to simplify any expression involving numbers, but we will now turn to a more fundamental issue, which is: why do the rules of algebra work at all? For instance, why is it true that a(b+c) is equal to ab + ac for any three numbers a, b, c? This is not an arbitrary choice of rule; it can be proven from more primitive, and more fundamental, properties of the number system.
Terence Tao

### Chapter 3. Set theory

Abstarct
Modern analysis, like most of modern mathematics, is concerned with numbers, sets, and geometry. We have already introduced one type of number system, the natural numbers.
Terence Tao

### Chapter 4. Integers and Rationals

Abstract
In Chapter 2 we built up most of the basic properties of the natural number system, but we have reached the limits of what one can do with just addition and multiplication. We would now like to introduce a new operation, that of subtraction, but to do that properly we will have to pass from the natural number system to a larger number system, that of the integers.
Terence Tao

### Chapter 5. The real numbers

Abstract
To review our progress to date, we have rigorously constructed three fundamental number systems: the natural number system N, the integers Z, and the rationals Q 1. We defined the natural numbers using the five Peano axioms, and postulated that such a number system existed; this is plausible, since the natural numbers correspond to the very intuitive and fundamental notion of sequential counting. Using that number system one could then recursively define addition and multiplication, and verify that they obeyed the usual laws of algebra.
Terence Tao

### Chapter 6. Limits of sequences

Abstract
In the previous chapter, we defined the real numbers as formal limits of rational (Cauchy) sequences, and we then defined various operations on the real numbers. However, unlike our work in constructing the integers (where we eventually replaced formal differences with actual differences) and rationals (where we eventually replaced formal quotients with actual quotients), we never really finished the job of constructing the real numbers, because we never got around to replacing formal limits LIM n→∞ a n with actual limits lim n→∞ a n . In fact, we haven’t defined limits at all yet. This will now be rectified.
Terence Tao

### Chapter 7. Series

Abstract
Now that we have developed a reasonable theory of limits of sequences, we will use that theory to develop a theory of infinite series.
Terence Tao

### Chapter 8. Infinite sets

Abstract
We now return to the study of set theory, and specifically to the study of cardinality of sets which are infinite (i.e., sets which do not have cardinality n for any natural number n), a topic which was initiated in Section 3.6.
Terence Tao

### Chapter 9. Continuous functions on R

Abstract
In previous chapters we have been focusing primarily on sequences. A sequence $$\left( {a_{n} } \right)_{n = 0}^{\infty }$$ can be viewed as a function from N to R, i.e., an object which assigns a real number a n to each natural number n. We then did various things with these functions from N to R, such as take their limit at infinity (if the function was convergent), or form suprema, infima, etc., or computed the sum of all the elements in the sequence (again, assuming the series was convergent).
Terence Tao

### Chapter 10. Differentiation of functions

Abstract
We can now begin the rigorous treatment of calculus in earnest, starting with the notion of a derivative. We can now define derivatives analytically, using limits, in contrast to the geometric definition of derivatives, which uses tangents. The advantage of working analytically is that (a) we do not need to know the axioms of geometry, and (b) these definitions can be modified to handle functions of several variables, or functions whose values are vectors instead of scalar. Furthermore, one’s geometric intuition becomes difficult to rely on once one has more than three dimensions in play. (Conversely, one can use one’s experience in analytic rigour to extend one’s geometric intuition to such abstract settings; as mentioned earlier, the two viewpoints complement rather than oppose each other.)
Terence Tao

### Chapter 11. The Riemann integral

Abstract
In the previous chapter we reviewed differentiation—one of the two pillars of single variable calculus. The other pillar is, of course, integration, which is the focus of the current chapter. More precisely, we will turn to the definite integral, the integral of a function on a fixed interval, as opposed to the indefinite integral, otherwise known as the antiderivative. These two are of course linked by the Fundamental theorem of calculus, of which more will be said later.
Terence Tao

### Backmatter

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