ANALYSIS LECTURES Contents 1. Sets 1 2. the Real Numbers 10 3

ANALYSIS LECTURES JOHN QUIGG Contents 1. Sets 1 2. The real numbers 10 3. Countability 21 4. Sequences 23 5. Limits 34 6. Continuity 38 7. Differentiation 42 8. Integration 48 9. Log and exp 60 10. Series 63 11. Sequences of functions 69 12. Series of functions 72 13. Power series 73 Index 80 1. Sets The terms set and element will be undefined. Remark 1.1. All a set A knows about is what its elements are. If x is any object, the sentence \x is an element of A", written \x A", is a (mathematical) proposition|that is, it is either true or false. Roughly2 speaking, all a set is good for is to allow the formation of such propositions. In fact, strictly speaking these are the only propositions in mathematics; more pre- cisely, every mathematical proposition can be built up from those of the form \x A". The rules governing these propositions comprise set theory, which is the2 foundation of modern mathematics. We won't study the formal theory of sets here; instead, we'll just record a few of the most important facts concerning sets, partly to indicate the level of rigor, and the style of writing, of our definitions, results (theorems, corollaries, propositions, and lemmas), and proofs. Virtually everything in this section, except perhaps for the general concept of families of sets occurring at the end, is prerequisite material for the course. Date: September 5, 2001. 1 2 JOHN QUIGG Many definitions in set theory are merely translations of constructions from logic; for example, the logical biconditional becomes set equality: Definition 1.2. Let A and B be sets. We say A equals B, written A = B, if for all x we have x A if and only if x B. 2 2 Notation and Terminology 1.3. The negation of \x A" is written \x A". 2 62 Definition 1.4. The empty set is the unique set such that for all x we have x . ; 62 ; Remark 1.5. There is no set A such that for all x we have x A; such a set would be \too big to be"|that is, its existence would lead2 to (logical) contradictions, of which the most famous is Russell's Paradox. The set-theoretic translation of the logical conditional is the subset rela- tion: Definition 1.6. Let A and B be sets. (i) A is called a subset of B, written A B, if for all x we have x A implies x B. ⊆ 2 (ii) If A B,2 we also say B is a superset of A, written B A. (iii) A subset⊆ A of B is called proper if A = B. ⊇ 6 Remark 1.7. In the above definition, we used a phrase of the form \we also say . ". It is important to realize that this just introduces a synonym for the primary term|the definition of each synonym is the same as that of the primary term. For example, the definition of \B A" is the same as that of \A B". ⊇ ⊆ Example 1.8. For every set A, we have: (i) A A; (ii) ⊆A. ; ⊆ Lemma 1.9. Let A and B be sets. Then A = B if and only if both A B and B A. ⊆ ⊆ Proof. First assume A = B. Since A A, we have both A B and B A. Conversely, assume A B and B ⊆A. Let x be any object.⊆ First assume⊆ x A. Then x B since⊆A B. Conversely,⊆ assume x B. Then x A since2 B A. Thus2 x A if and⊆ only if x B, so A = B.2 2 ⊆ 2 2 Remark 1.10. In the forward direction of the above proof, we had to prove A B (as well as B A); a more typical proof of this would start with \let⊆x A" and would⊆ deduce \x B", however, the hypothesis \A = B" was so2 special that it trivially implied2 the desired conclusion. Actually, the above lemma has a much shorter proof, because it is just a set-theoretic version of a tautology from logic (namely, the biconditional p q is equivalent to the conjunction p q and q p). However, the whole, point here is to indicate typical proofs) with sets,) not to give the most efficient proofs of the results themselves. ANALYSIS LECTURES 3 The set-theoretic version of the logical disjunction, conjunction, and negation are union, intersection, and complement: Definition 1.11. Let A an B be sets. (i) The union of A and B is defined by A B := x : x A or x B : [ f 2 2 g (ii) The intersection of A and B is defined by A B := x : x A and x B : \ f 2 2 g (iii) A and B are called disjoint if A B = . (iv) The difference of A and B is defined\ by; A B := x : x A and x B : n f 2 62 g (v) If the set U is understood, for any subset A U the complement of A is defined by ⊆ Ac := U A: n Remark 1.12. Rather than state the elementary properties of these opera- tions, we'll wait until we have the more general versions for families of sets (set below). Note that the complement of A is not just x : x A , rather it is x U : x A . Thus the complement dependsf upon62 theg choice of the particularf 2 \universe"62 g U; there is no single set U that could serve as a universe for taking complements of every set A, because, as we mentioned above, there is no single set U which contains all sets. Definition 1.13. (i) Let x and y be objects. The ordered pair with first coordinate x and second coordinate y is defined by (x; y) := x ; x; y : f g f g (ii) The Cartesian product of sets A and B is defined by A B := (x; y): x A and x B : × f 2 2 g Remark 1.14. Note that ordered pairs (x; y) and (z; w) are equal if and only if x = z and y = w. However, it would be improper to try to define \ordered pair" by this property; it is important to know that we can define all our concepts completely in terms of sets. Notation and Terminology 1.15. R denotes the set of real numbers, which we will discuss in more detail in the next section. Definition 1.16. Let A and B be sets, and let f A B. We say f is a function from A to B, written f : A B, if for⊆ all x× A there exists a unique y B such that (x; y) f. We! call this y the2value of f at x, written y =2f(x). 2 Remark 1.17. It would be improper to try to define a \function" as a \rule" with certain properties, because it raises the question \what is a rule?"; again, it is important to know that we can phrase the definition in terms of sets. 4 JOHN QUIGG Definition 1.18. Let f : A B. ! (i) The set A is called the domain of f, written A = dom f. (ii) The range of f is defined by ran f := f(x): x A : f 2 g (iii) If C A, the image of C under f is defined by ⊆ f(C) := f(x): x C : f 2 g (iv) If D B, the pre-image of D under f is defined by ⊆ 1 f − (D) := x A : f(x) D : f 2 2 g (v) f is called real-valued if ran f R. ⊆ Remark 1.19. Note that the set A is determined by the function f, since A = dom f. However, the set B is not determined by f|it could be any superset of the range of f. This has an important consequence: if we have a function f : A B, and if C is any set such that ran f C, then we can equally well! regard f : A C; it is still the same function.⊆ Actually, we have made a choice here|in! some areas of advanced mathematics it is important to regard the set B as an official part of the function, but we have chosen to identify the function with its graph (x; f(x)) : x dom f . f 2 g Remark 1.20. The notation f(C) for images is an abuse of the function notation; we must take care to interpret the notation correctly from the 1 context. Even more dangerous is the notation f − (D) for pre-images, since it can be confused with the concept of an inverse function (see below). Notation and Terminology 1.21. Occasionally it is convenient to intro- duce a function but not give it a name, in which case we use the notation \x (some formula involving x)". For example, we could consider \the function7! x x2 from R to R". 7! Remark 1.22. Thus, a function f is just a set of ordered pairs with a certain property, popularly known as the \vertical line test": for all x; y; z, if (x; y) f and (x; z) f, then y = z. In particular, to prove two functions are equal2 is just a case2 of proving two sets are equal. But since functions are such special kinds of sets, there is a more efficient way to prove function equality: Lemma 1.23. Let f and g be functions.

Load more