Many of these ideas are, on a conceptual or practical level, dealt with at lower levels of mathematics, including a regular First-Year Calculus course, and so, to the uninitiated reader, the subject of Real Analysis may seem rather senseless and trivial. However, Real Analysis is at a depth, complexity, and arguably beauty, that it is because under the surface of everyday mathematics, there is an assurance of correctness, that we call rigor, that permeates the whole of mathematics.
Thus, Real Analysis can, to some degree, be viewed as a development of a rigorous, well-proven framework to support the intuitive ideas that we frequently take for granted. Real Analysis is a very straightforward subject, in that it is simply a nearly linear development of mathematical ideas you have come across throughout your story of mathematics. However, instead of relying on sometimes uncertain intuition which we have all felt when we were solving a problem we did not understand , we will anchor it to a rigorous set of mathematical theorems.
Throughout this book, we will begin to see that we do not need intuition to understand mathematics - we need a manual. The overarching thesis of this book is how to define the real numbers axiomatically. How would that work?
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This book will read in this manner: we set down the properties which we think define the real numbers. We then prove from these properties - and these properties only - that the real numbers behave in the way which we have always imagined them to behave. We will then rework all our elementary theorems and facts we collected over our mathematical lives so that it all comes together, almost as if it always has been true before we analyzed it; that it was in fact rigorous all along - except that now we will know how it came to be.
Do not believe that once you have completed this book, mathematics is over. In other fields of academic study, there are glimpses of a strange realm of mathematics increasingly brought to the forefront of standard thought. After understanding this book, mathematics will now seem as though it is incomplete and lacking in concepts that maybe you have wondered before. In this book, we will provide glimpses of something more to mathematics than the real numbers and real analysis. After all, the mathematics we talk about here always seems to only involve one variable in a sea of numbers and operations and comparisons.
Note: A table of the math symbols used below and their definitions is available in the Appendix. A select list of chapters curated from other books are listed below. They should help develop your mathematical rigor that is a necessary mode of thought you will need in this book as well as in higher mathematics. This part of the book formalizes the various types of numbers we use in mathematics, up to the real numbers. This part focuses on the axiomatic properties what we have defined to be true for the sake of analysis of not just the numbers themselves but the arithmetic operations and the inequality comparators as well.
This part of the book formalizes the definition and usage of graphs, functions, as well as trigonometry. The most curious aspect of this section is its usage of graphics as a method of proof for certain properties, such as trigonometry. These methods of proof are mostly frowned upon due to the inaccuracy and lack of rigorous definition when it comes to graphical proofs , but they are essential to derive the trigonometric relationships, as the analytical definition of the trigonometric functions will make using trigonometry too difficult—especially if they are described early on.
The following chapters will rigorously define the trigonometric functions.
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Often, such proofs tend to be shorter or simpler compared to classical proofs that apply direct methods. A sequence is a function whose domain is a countable , totally ordered set. The domain is usually taken to be the natural numbers  , although it is occasionally convenient to also consider bidirectional sequences indexed by the set of all integers, including negative indices. A sequence that tends to a limit i. See the section on limits and convergence for details.
If either holds, the sequence is said to be monotonic. Roughly speaking, a limit is the value that a function or a sequence "approaches" as the input or index approaches some value. The idea of a limit is fundamental to calculus and mathematical analysis in general and its formal definition is used in turn to define notions like continuity , derivatives , and integrals.
In fact, the study of limiting behavior has been used as a characteristic that distinguishes calculus and mathematical analysis from other branches of mathematics. The concept of limit was informally introduced for functions by Newton and Leibniz , at the end of 17th century, for building infinitesimal calculus. We write this symbolically as.
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Sometimes, it is useful to conclude that a sequence converges, even though the value to which it converges is unknown or irrelevant. In these cases, the concept of a Cauchy sequence is useful. It can be shown that a real-valued sequence is Cauchy if and only if it is convergent. In a general metric space, however, a Cauchy sequence need not converge. In addition, for real-valued sequences that are monotonic, it can be shown that the sequence is bounded if and only if it is convergent.
However, in the case of sequences of functions, there are two kinds of convergence, known as pointwise convergence and uniform convergence , that need to be distinguished. In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely.
The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations e. For example, a sequence of continuous functions see below is guaranteed to converge to a continuous limiting function if the convergence is uniform, while the limiting function may not be continuous if convergence is only pointwise.
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Karl Weierstrass is generally credited for clearly defining the concept of uniform convergence and fully investigating its implications. Compactness is a concept from general topology that plays an important role in many of the theorems of real analysis. The property of compactness is a generalization of the notion of a set being closed and bounded. In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and bounded.
Briefly, a closed set contains all of its boundary points , while a set is bounded if there exists a real number such that the distance between any two points of the set is less than that number.
The equivalence of the definition with the definition of compactness based on subcovers, given later in this section, is known as the Heine-Borel theorem. A more general definition that applies to all metric spaces uses the notion of a subsequence see above.
This particular property is known as subsequential compactness. Subsequential compactness is equivalent to the definition of compactness based on subcovers for metric spaces, but not for topological spaces in general. Compact sets are well-behaved with respect to properties like convergence and continuity. For instance, any Cauchy sequence in a compact metric space is convergent.
As another example, the image of a compact metric space under a continuous map is also compact. A function from the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane ; such a function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps". There are several ways to make this intuition mathematically rigorous.
Several definitions of varying levels of generality can be given. In cases where two or more definitions are applicable, they are readily shown to be equivalent to one another, so the most convenient definition can be used to determine whether a given function is continuous or not. This definition, which extends beyond the scope of our discussion of real analysis, is given below for completeness.
On a compact set, it is easily shown that all continuous functions are uniformly continuous. The collection of all absolutely continuous functions on I is denoted AC I. Absolute continuity is an important concept in the Lebesgue theory of integration, allowing the formulation of a generalized version of the fundamental theorem of calculus that applies to the Lebesgue integral. The notion of the derivative of a function or differentiability originates from the concept of approximating a function near a given point using the "best" linear approximation.
If the derivative exists everywhere, the function is said to be differentiable.
Differentiability is therefore a stronger regularity condition condition describing the "smoothness" of a function than continuity, and it is possible for a function to be continuous on the entire real line but not differentiable anywhere see Weierstrass's nowhere differentiable continuous function. It is possible to discuss the existence of higher-order derivatives as well, by finding the derivative of a derivative function, and so on. One can classify functions by their differentiability class. A series formalizes the imprecise notion of taking the sum of an endless sequence of numbers.
The idea that taking the sum of an "infinite" number of terms can lead to a finite result was counterintuitive to the ancient Greeks and led to the formulation of a number of paradoxes by Zeno and other philosophers. The modern notion of assigning a value to a series avoids dealing with the ill-defined notion of adding an "infinite" number of terms. The series is assigned the value of this limit, if it exists. It is to be emphasized that the word "sum" is used here in a metaphorical sense as a shorthand for taking the limit of a sequence of partial sums and should not be interpreted as simply "adding" an infinite number of terms.
For instance, in contrast to the behavior of finite sums, rearranging the terms of an infinite series may result in convergence to a different number see the article on the Riemann rearrangement theorem for further discussion. An example of a convergent series is a geometric series which forms the basis of one of Zeno's famous paradoxes :.
In contrast, the harmonic series has been known since the Middle Ages to be a divergent series:.
It is easily shown that absolute convergence of a series implies its convergence. On the other hand, an example of a conditionally convergent series is. Even a converging Taylor series may converge to a value different from the value of the function at that point. If the Taylor series at a point has a nonzero radius of convergence , and sums to the function in the disc of convergence , then the function is analytic.