Introduction to Analysis

By Maxwell Rosenlicht

This well-written text provides excellent instruction in basic real analysis, giving a solid foundation for direct entry into advanced work in such fields as complex analysis, differential equations, integration theory, and general topology. The nominal prerequisite is a year of calculus, but actually nothing is assumed other than the axioms of the real number system. Because of its clarity, simplicity of exposition, and stress on easier examples, this material is accessible to a wide range of students, of both mathematics and other fields.
Chapter headings include notions from set theory, the real number system, metric spaces, continuous functions, differentiation, Riemann integration, interchange of limit operations, the method of successive approximations, partial differentiation, and multiple integrals.
Following some introductory material on very basic set theory and the deduction of the most important properties of the real number system from its axioms, Professor Rosenlicht gets to the heart of the book: a rigorous and carefully presented discussion of metric spaces and continuous functions, including such topics as open and closed sets, limits and continuity, and convergent sequence of points and of functions. Subsequent chapters cover smoothly and efficiently the relevant aspects of elementary calculus together with several somewhat more advanced subjects, such as multivariable calculus and existence theorems. The exercises include both easy problems and more difficult ones, interesting examples and counter examples, and a number of more advanced results.
Introduction to Analysis lends itself to a one- or two-quarter or one-semester course at the undergraduate level. It grew out of a course given at Berkeley since 1960. Refinement through extensive classroom use and the author’s pedagogical experience and expertise make it an unusually accessible introductory text.

• Language: English
• Publisher: Dover Publications
• Release date: May 4, 2012
• ISBN: 9780486134680

Book preview

Analysis

CHAPTER I

Notions from Set Theory

Set theory is the language of mathematics. The most complicated ideas in modern mathematics are developed in terms of the basic notions of set theory. Fortunately the grammar and vocabulary of set theory are extremely simple, at least in the sense that it is possible to go very far in mathematics with only a small amount of set theory. It so happens that the subject of set theory not only underlies mathematics but has become itself an extensive branch of study; however we do not enter deeply into this study because there is no need to. All we must do here is familiarize ourselves with some of the basic ideas so that the language may be used with precision. A first reading of this chapter can be very rapid since it is mainly a matter of getting used to a few words. There is occasional verbosity, directed toward the clarification of certain simple ideas which are really somewhat more subtle than they appear.

§ 1. SETS AND ELEMENTS. SUBSETS.

We do not attempt to define the word set. Intuitively a set is a collection, or aggregate, or family, or ensemble (all of which words are used synonymously with set) of objects which are called the elements, or members of the set, and the set is completely determined by the knowledge of which objects are elements of it. We may speak, for example, of the set of students at a certain university; the elements of this set are the individual students there. Similarly we may speak of the set of all real numbers (to be discussed in some detail in the next chapter), or the set of all straight lines in a given plane, etc. It should be noted that the elements of a set may themselves be sets; for example each element of the set of all straight lines in a given plane is a set of points, and we may also consider such less mathematical examples as the set of married couples in a given town, or the set of regiments in an army.

We shall generally use capital letters to denote sets and lower-case letters to denote their elements. The symbol ∈ is used to denote membership in a set, so that

means that x is an element of the set S. The statement "x is not an element of S" is abbreviated

Instead of writing a ∈ S, b ∈ S, c ∈ S (the commas having the same meaning as and) we often write a, b, c ∈ S.

The statement that a set is completely determined by its elements may be written as follows: If X and Y are sets then X = Y if and only if, for all x, x ∈ X if and only if x ∈ Y. Equality here and elsewhere in this book (denoted by =) means identity; X and Y happen to be different symbols, but they may very well be different names for the same set, in which case the equation X = Y means that the sets indicated by the symbols X and Y are the same. X ≠ Y of course means that the sets indicated by the symbols X and Y are not the same.

Thus we imagine ourselves in a world peopled by certain objects (certain of which are called sets), and for some pairs of objects x, X, where X is a set, we write x ∈ X, the symbol ∈ having the property that two sets X and Y are equal if and only if for each object x we have x ∈ X if and only if x ∈ Y. The symbol ∈ must also have other properties (which we don’t specify here) that enable us, given certain sets, to construct others. The important thing is that everything which follows is expressible in terms of the fundamental relation x ∈ X.

Sets are sometimes indicated by listing all their members between braces. For example, the set

{Jane, Jim}

has Jane and Jim as its members,

{1, 2, 3, …}

(with the three dots read and so on) is the set of positive integers, and

{a}

is the set having one element, the object named a. (Note that {a} is not the same as a. In the same way there is a difference between a university class consisting of one student and the student himself, or between a committee consisting of one person and that person.) The above notation however is not always feasible. A more frequently used notation is

{x : (statement involving x) },

which means the set of all x for which the statement involving x is true. Thus

{x ; x is a positive integer}

is the set of positive integers, and for any set S we have

For any set S, the symbol

denotes the same set as

which is the set of all elements of S for which the statement is true. Thus if R is the set of real numbers,

If X and Y are sets and every element of X is also an element of Y, we say that X is a subset of Y; this is written

X ⊂ Y or Y ⊂ X

Thus X ⊂ Y is shorthand for the statement "if x ∈ X then x ∈ X". X = Y is equivalent to the two statements X ⊂ Y and Y ⊂ X. If X ⊂ Y and Y ⊂ Z then clearly X ⊂ Z; the two first statements are sometimes written more succinctly

For any object x and set X the relation x ∈ X can now be written in another (less convenient!) way, namely {x} ⊂ X. The negation of X ⊂ Y is written

X is called a proper subset of Y if X ∈ Y but X ≠ Y.

The empty set is the set with no elements. It is denoted by the symbol Ø. A source of confusion to beginners is that although the empty set contains nothing, it itself is something (namely some particular set, the one characterized by the fact that nothing is in it). The set {Ø} is a set containing exactly one element, namely the empty set. (In a similar way, when dealing with numbers, say with ordinary integers, we must be careful not to regard the number zero as nothing: zero is something, a particular number, which represents the number of things in nothing. Thus zero and Ø are quite different, but there is a connection between them in that the set Ø has zero elements.) Note that for any set X we have

A special case of both of these statements is the statement

which occasions difficulty if, as is often improperly done, one reads is contained in for both of the symbols ⊂ and ∈. The statement Ø ⊂ Ø is true because the statement "for each x ∈ Ø we have x ∈ Ø is obviously true, and also because it is vacuously true", that is there is no x ∈ Ø for which the statement must be verified, just as the statement all pigs with wings speak Chinese is vacuously true.

§ 2. OPERATIONS ON SETS.

If X and Y are sets, the intersection of X and Y, denoted by X ∩ Y, is defined to be the set of all objects which are both elements of X and elements of Y. In symbols,

The union of X and Y, denoted X ∈Y, is the set of all objects which are elements of at least one of the sets X and Y. That is. X ∈Y is the set of all objects which are either elements of X or elements of Y (or of both), in symbols

The word or is used here in the manner that is standard in mathematics. In ordinary language the word or is often exclusive, that is, if A and B are statements, then "A or B is understood to mean A or B but not both", whereas in mathematics it always means "A or B, or both A and B".

If X is a subset of a set S, then the complement of X in S is the set of all elements of S which are not elements of X. If it is explicitly stated, or clear from the context, exactly what the set S is, we often omit the words "in S" for the complement of X. That is

These operations are illustrated in Figure 1, where the sets in question are sets of points in plane regions bounded by curves.

FIGURE 1.Intersection, union, and complement.

For another example, let S be the set of real numbers, and let

(The symbols ≥ and ≤ will be defined later.) Then

. For example, if X and Y are subsets of a set S, then

This is illustrated in Figure 2. A proof of this formula is given below.

.

EXERCISE. Prove that if X ⊂ S, Y ⊂ S.

It must be shown that the two sets have the same elements, in other words that each element of the set on the left is an element of the set on the right and vice versa.

. This means that x ∈ S, x ∉ X, x ∉ Y Since x ∉ X, x ∉ Y.

, then x ∈ S . Therefore x ∉ X and x ∉ Y. This completes the proof.

If X and Y are sets, the notation X – Y . Thus if X and Y are subsets of some set S, .

Two sets are said to be disjoint if they have no element in common. That is, X and Y are disjoint if X ∩ Y = Ø. A collection of any number of sets is said to be disjoint if every two of the sets are disjoint.

The intersection and union of more than two sets may be defined in an obvious manner. For example, if X, Y, Z are sets then

and

Clearly

, and similarly for the union of three sets. More generally the intersection and union of arbitrary families of sets may be defined, and in an obvious way. The only problem is finding an adequate notation for an arbitrary family of sets, and this is done as follows. Let I be any set and for each i ∈ I let Xi be another set (so that we may speak of I as being an indexing family, whose elements are indices used to specify the sets at which we direct our main attention). The set of all sets Xi as i ranges over I is denoted

and the intersection and union of this family of sets, together with their respective conventional symbols, are defined by

EXERCISE. Prove that if I and S are sets and if for each i ∈ I we have Xi ⊂ S,

It must be shown that each element of the set on the left is an element of the set on the right, and vice versa.

then x ∈ S , for at least one j ∈ I. .

, then for some j ∈ I . Thus x ∈ S . . . This completes the proof.

If a and b are objects, by the ordered pair (a, b) we mean the two objects a and b in a definite order, a first, b second. Thus if a, b, c, d are objects then (a, b) = (c, d) if and only if a = c and b = d. Note the distinction between (a, b) and {a, b}; the latter is a set with two elements (unless, of course, a happens to equal b, in which case {a, b} = {a}, a set with one element), and {a, b} can equally well be written {b, a}, spoiling the order. We remark that instead of introducing the new concept ordered pair into set theory, we can actually define the ordered pair (a, b) in terms of the primitive notions about sets that we already have: we set (a, b) = {{a}, {a, b}}. This definition does precisely what we want: to any two objects a, b (distinct or not) it assigns an object (a, b), and it does this in such a fashion that (a, b) = (c, d) if and only if a = c and b = d.

Given two sets X and Y we define the cartesian product (or product) of X and Y, denoted X × Y, to be the set of all ordered pairs the first member of which is in X, the second in Y, that is

Ordinary rectangular coordinates in the plane give the usual pictorial representation of the cartesian product: the whole plane can be identified with the product of the two coordinate axes. In Figure 3 there is a more complicated picture in which X, Y are subsets of the two coordinate axes and the cartesian product is a subset of the first quadrant.

FIGURE 3.Cartesian product.

§ 3. FUNCTIONS.

If X and Y are sets, by a function from X to Y (or a function from X into Y, or a function on X with values in Y) is meant a rule which associates with each element of X a definite element of Y. (The word mapping, or map, is often used instead of function.) The rule can be given in many ways, some of which are discussed below, but the essential thing is that given any element of X there is associated, somehow, some definite element of Y. Two functions from X to Y are considered equal if and only if both functions associate with each specific element of X the same element of Y.

Functions are usually denoted by small letters, such as f. The statement "f is a function from X to Y" is often written

f : X → Y.

For any x ∈ X, the element of Y that the function f associates with x (the value of f at x) is denoted f (x). Thus if f: X → Y and g: X → Y then f = g if and only if f(x) = g(x) for all x ∈ X. We say that f sends x into f(x), or that f maps x into f(x), or that x and f(x) correspond under f.

The rule defining a given function f: X → Y may be given in various ways. One way, which is usually not very practical, is to list all the elements of X, listing with each one the corresponding one of Y. Or the rule may be given by a mathematical formula. For example, if X and Y are both taken to be the set R of real numbers, an equation like

f(x) = x³ + 3x – 2

defines a real-valued function f on R ; in such a case one often speaks (imprecisely!) of the function x³ + 3x – 2. Again, if X is a subset of R, a real-valued function on X may be given geometrically by its graph, that is the set of points { (x, f(x)) : x ∈ X} in the plane; note that this method may or may not be practical, depending on what f is like, for it may not be possible to draw the graph. In fact any subset of the plane defines a real-valued function on a subset of the real numbers, provided that any vertical line (x = constant) intersects the subset of the plane in at most one point. Finally we remark that the rule defining a function need not be practically computable. For example, for x any real number, let f(x) denote that integer 0,1, …, 9 which is in the billionth decimal place of x (to be precise, since a real number x may have more than one decimal representation, as in 1.0000… = .9999…, we might better take f(x) to be the smallest possible integer in the billionth decimal place of x); this rule gives an honest-to-goodness function f: R → R, but who would hazard a guess as to the value of f(π),

Though this is in no way essential for what follows, we remark that it is easy to define the notion of function in terms of more primitive concepts of set theory, as follows: If X and Y are sets, a function from X into Y is a subset of X × Y with the property that for any x ∈ X there is one and only one y ∈ Y such that (x, y) is in the subset. If the function is denoted f: X → Y then the unique y referred to above is, of course, f(x). In analogy with the case of real-valued functions on the real numbers, this subset {(x, f(x)) : x ∈ X} of X × Y is called the graph of the function, so that we are again saying that the graph determines the function.

FIGURE 4.Graph of the function f: R → R given by f(x) = x² for all x ∈ R.

It is useful to note that the word function alone can be defined in primitive terms, not only the more complete concept "function from X into Y": a function is an ordered pair whose first member is an ordered pair of sets, say (X, Y), and whose second member is a function from X into Y. That is, a function is something of the type ((X, Y), (a certain kind of subset of X × Y)). This emphasizes that the sets X and Y are to be considered as essential parts of the function f: X → Y. For many purposes it is important to bear this fact in mind, but most often we do not make any explicit mental note of it. For example, if f: X → Y is a function and Y is a subset of another set Y′, then we get a function f′: X → Y′ by setting f′(x) = f(x) for all x ∈ X, that is, giving f′ the same graph as f, which is possible since X × Y ⊂ X × Y′. Although f and f′ are really different functions, we usually denote them by the same symbol, even writing down the technically incorrect expression f: X → Y′. In the same way, given any function f: X → Y and any subset X ″ of X, we can define a function f″: X″ → Y by f″(x) = f(x) for all x ∈ X″; f″ is called the restriction of f to X″. Of course, f″ is not the same as f, though we often denote them by the same symbol, for example by writing the technically incorrect expression f: X″ → Y when there is no danger of confusion. The graph of the latter function is of course a subset of the graph of the original function f: X→ Y, since X″ × Y ⊂ X × Y.

If f: X → Y and g: Y → Z are functions, one can define the composition of f and g, or composed function, a function from X into Z, by associating to each element of X an element of Z in the obvious way: given an element of X, one first uses f to get an element of Y, then one uses g to get from this last element an element of Z. The composed function is usually denoted g o f, so that we have

for each x ∈ X.

A function f: X → Y is called one-to-one, or one-one, if different elements of X correspond under f to different elements of Y, that is if f (x1) = f(x2) only if x1 = x2. A function f: X → Y is called onto if each element of Y corresponds under f to some element of X, that is if each y ∈ Y is of the form y = f(x), for some x ∈ X. If f: X → Y is both one-one and onto it is called one-one onto, or a one-one correspondence between X and Y. An example of a function that is one-one onto is, for any set X, the identity function iX: X → X, given by iX(x) = x for all x ∈ X. (Note that what goes under the name "the function x" in elementary calculus is actually the identity function on the set of real numbers.) If f: X → Y is one-one onto then each element of Y corresponds under f to one and only one element of X, so we can define a function f−1: Y → X by f−1(y) = x if y = f(x). f−1 is called the inverse function of f, and is also one-one onto. Clearly (f−1)−1 = f.

If f: X → Y , then the subset of Y given by

(where the last symbol is shorthand for {y : there exists x ∈ X′ such that y = f(x)}) is called the image of X′ under f, or simply the image of X′, if there is no danger of confusion. The two uses we have made for the symbol f( ) are related by the equation

If f: X → Y , then the subset of X given by

is called the inverse image of Y′ under f. It consists of those elements of X which correspond under f to elements of Y′. If f: X → Y happens to be one-one onto we have another use for the symbol f−1, namely the inverse function f−1: Y → X. These two uses of f−1 must be carefully distinguished, though confusion rarely arises. If f: X → Y is one-one onto then the two uses of ffor each y ∈ Y.

§ 4. FINITE AND INFINITE SETS.

We are familiar with the set of positive integers, or natural numbers {1, 2, 3, …}. This set, together with the various ideas associated with it, such as its ordering (the fact that its elements can be written down in a definite order), or such as the fact that two of its elements may be added to obtain a third with certain general rules holding for this addition, can be obtained from the primitive principles of set theory. In this text we shall instead assume the basic properties of the real number system and from those derive all the properties of the set {1, 2, 3, …}. In this section we shall for convenience assume a few simple facts about the natural numbers in order to get as quickly as possible to certain other easy matters of set theory. However all the facts about the set of natural numbers that are used here will be proved explicitly in the next chapter. The notions developed in this section will not be applied until later, so no circular reasoning occurs.

Let us therefore assume knowledge of the set {1, 2, 3, …}. A set X is called finite if it is empty or there is a positive integer n such that X can be put into one-one correspondence with the set {1,2, …, n}, that is there is a one-one function from {1,2, …, n) onto X. Thus a set is finite if we can count its elements and run out of elements after we count a certain number, say n, of them. The number n depends only on the set X, not on the order in which its elements are counted off; n is called the number of elements in X. For Completeness we say that the number of elements in the empty set is zero. Any subset of a finite set is itself finite, and if it is a proper subset it has a smaller number of elements.

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