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INTERMEDIATE CALCULUS

AND

LINEAR ALGEBRA

Part I

J. KAZDAN

Harvard University Lecture Notes

ii

Preface

These notes will contain most of the material covered in class, and be distributed before each lecture (hopefully). Since the course is an experimental one and the notes written before the lectures are delivered, there will inevitably be some sloppiness, disorganization, and even egregious blunders—not to mention the issue of clarity in exposition. But we will try. Part of your task is, in fact, to catch and point out these rough spots. In mathematics, proofs are not dogma given by authority; rather a proof is a way of convincing one of the validity of a statement. If, after a reasonable attempt, you are not convinced, complain loudly. Our subject matter is intermediate calculus and linear algebra. We shall develop the material of linear algebra and use it as setting for the relevant material of intermediate calculus. The first portion of our work—Chapter 1 on infinite series—more properly belongs in the first year, but is relegated to the second year by circumstance. Presumably this topic will eventually take its more proper place in the first year. Our course will have a tendency to swallow whole two other more advanced courses, and consequently, like the duck in Peter and the Wolf, remain undigested until regurgitated alive and kicking. To mitigate—if not avoid—this problem, we shall often take pains to state a theorem clearly and then either prove only some special case, or offer no proof at all. This will be true especially if the proof involves technical details which do not help illuminate the landscape. More often than not, when we only prove a special case, the proof in the general case is essentially identical—the equations only becoming larger. September 1964

iv

to the reals. My current feeling is to consider linear and non-linear maps between finite dimensional spaces before doing the infinite dimensional example of differential equations. The first semester should get up to the generalities on solving LX = Y , p. 319 [incidentally, the material on inverses (p. 355 ff) belongs around p. 319]. Most students find the material on linear dependence difficult—probably for two reasons:

1. they are not used to formal definitions, and ii) they think they have learned a technique for doing something, not just a naked definition, and can’t quite figure out just what they can do with it. In other words, they should feel these definitions about the anatomy of linear spaces are similar to those describing a football field and of little value until the game begins—i.e., until the operators between spaces make their grand entrance. Because of time shortages, the sections on linear maps from R^1 → Rn^ and Rn^ → R^1 , pp. 320-41 were regrettably omitted both years I taught the course. The notes were written so that these sections can be skipped.

(C) Supplementary Material. A remarkable number of fascinating and important topics could have been included—if there were only enough time. For example:

(1) Change of bases for linear transformations (including the spectral theorem). (2) Elementary differential geometry of curves and surfaces. (3) Inverse and implicit function theorems. These should be stated as natural gener- alizations of the problems of a) inverting a linear map, b) finding the null space of a linear map, and c) generalizing dim D(L) = dim R(L) + dim N (L) all to local properties of nonlinear maps via the tangent map. (4) Change of variable in multiple integration. Determinants were deliberately in- troduced as oriented volume to make the result obvious for linear maps and plausible for nonlinear maps. (5) Constrained extrema using Lagrange multipliers. (6) Line and surface integrals along with the theorems of Gauss, Green, and Stokes. The formal development of differential forms takes too much time to do here. Perhaps a satisfactory solution is to restrict oneself to line integrals and these theorems in the plane, where the topological difficulties are minimal. (7) Elementary Morse Theory. One can prove the Morse inequalities easily for the real line, the circle, the plane, and S^2 merely by gradually flooding these sets and observing the number of lakes and shore line changes only at the critical points. (8) Sturm-Liouville theory. An elegant fusion of the geometry of Hilbert spaces to differential equations. (9) Translation-invariant operators with applications to constant coefficient differ- ence and differential equations. The Laplace and Fourier transforms enter natu- rally here. (10) The Calculus of Variations. The formalism of nonlinear functionals on Rn^ , i.e., maps f : Rn^ → R , generalizes immediately to nonlinear functionals defined on infinite dimensional spaces.

v

(11) The deleted rigor. (12) Linear operators with finite dimensional (perhaps even compact) range.

One parting warning. When covering intermediate calculus from this viewpoint, it is all too natural to forget the innocence of the class, to enchant with glitter, and to numb with purity and formalism. Emphasis should be placed on developing insight and intuition along with routine computational facility. My classes found frequent reviews of the mathematical edifice, backward glances at the previous months’ work, not only helpful but mandatory if they were to have any conception of the vast canvas which was being etched in their minds over the course of the year. The question, “What are we doing now and how does it fit into the larger plan?” must constantly be raised and at least partially resolved. May, 1966

Contents

• 0 Remembrance of Things Past.
  • 0.1 Sets and Functions
  • 0.2 Relations
  • 0.3 Mathematical Induction
  • 0.4 Reals: Algebraic and Order Properties
  • 0.5 Reals: Completeness
  • 0.6 Appendix: Continuous Functions and the Mean Value Theorem
  • 0.7 Complex Numbers: Algebraic Properties
  • 0.8 Complex numbers: Completeness and Functions
• 1 Infinite Series
  • 1.1 Introduction
  • 1.2 Tests for Convergence of Positive Series
  • 1.3 Absolute and Conditional Convergence
  • 1.4 Power Series, Infinite Series of Functions
  • 1.5 Properties of Functions Represented by Power Series
  • 1.6 Complex-Valued Functions, ez , cos z, sin z
  • 1.7 Appendix to Chapter 1, Section 7.
• 2 Linear Vector Spaces: Algebraic Structure
  • 2.1 Examples and Definition
    • a) The Space R
    • b) The Space Rn
    • c) The Space C[a, b]
    • d) D. The Space Ck[a, b]
    • e) E. The Space l
    • f) F. The Space L 1 [a, b]
    • g) G. The Space fn
    • h) Appendix. Free Vectors
  • 2.2 Subspaces. Cosets.
  • 2.3 Linear Dependence and Independence. Span.
  • 2.4 Bases and Dimension
• 3 Linear Spaces: Norms and Inner Products
  • 3.1 Metric and Normed Spaces
  • 3.2 The Scalar Product in E
  • 3.3 Abstract Scalar Product Spaces
  • 3.4 Fourier Series. viii CONTENTS
  • 3.5 Appendix. The Weierstrass Approximation Theorem
  • 3.6 The Vector Product in R
• 4 Linear Operators: Generalities. V 1 → Vn, Vn → V
  • 4.1 Introduction. Algebra of Operators
  • 4.2 A Digression to Consider au′′ + bu′ + cu = f
  • 4.3 Generalities on LX = Y
  • 4.4 L : R^1 → Rn Parametrized Straight Lines.
  • 4.5 L : Rn → R^1 Hyperplanes.
• 5 Matrix Representation
  • 5.1 L : Rm → Rn
  • 5.2 Supplement on Quadratic Forms
  • 5.3 Volume, Determinants, and Linear Algebraic Equations.
    • a) Application to Linear Equations
  • 5.4 An Application to Genetics
  • 5.5 A pause to find out where we are
• 6 Linear Ordinary Differential Equations
  • 6.1 Introduction
  • 6.2 First Order Linear
  • 6.3 Linear Equations of Second Order
    • a) A Review of the Constant Coefficient Case.
    • b) Power Series Solutions
    • c) General Theory
  • 6.4 First Order Linear Systems
  • 6.5 Translation Invariant Linear Operators
  • 6.6 A Linear Triatomic Molecule
• 7 Nonlinear Operators: Introduction
  • 7.1 Mappings from R^1 to R^1 , a Review
  • 7.2 Generalities on Mappings from Rn to Rm
  • 7.3 Mapping from E^1 to En
• 8 Mappings from En to E : The Differential Calculus
  • 8.1 The Directional and Total Derivatives
  • 8.2 The Mean Value Theorem. Local Extrema.
  • 8.3 The Vibrating String.
    • a) The Mathematical Model
    • b) Uniqueness
    • c) Existence
  • 8.4 Multiple Integrals
• 9 Differential Calculus of Maps from En to Em , s.
  • 9.1 The Derivative
  • 9.2 The Derivative of Composite Maps (“The Chain Rule”).
• 10 Miscellaneous Supplementary Problems

Chapter 0

Remembrance of Things Past.

We shall treat a hodge-podge of topics in a hasty and incomplete fashion. While most of these topics should have been learned earlier, section 5 on the completeness of the real numbers has its more rightful place in advanced calculus. Do not take time to read this chapter unless the particular topic is needed; then read only the relevant portions. The chapter is included for reference.

0.1 Sets and Functions

A set is any collection of objects, called the elements of the set, together with a criterion for deciding if an object is in the set. For example, I) the set of all girls with blue eyes and blond hair, and ii) the less picturesque set of all positive even integers. We can also define a set by bluntly listing all of its elements. Thus, the set of all students in this class is defined by the list in the roll book.

Sets are often specified by a notation which is best described by examples. i) S = { x : x is an integer } is the set of all integers. ii) T = { (x, y) : x^2 +y^2 = 1 } is the set of all points (x, y) on the unit circle x^2 +y^2 = 1. iii) A = { 1 , 2 , 7 , − 3 } is the set of integers 1, 2 , 7 and −. Our attitude toward set theory will be extremely casual; we shall mainly use it as a language and notation. Without further ado, let us introduce some notation. x ∈ S, x is an element of the set S , or just x is in S. x 6 ∈ S, x is not an element of the set S. Z, the set of all integers, positive, zero, and negative. Z+, the set of all positive integers, excluding 0. R the set of all real numbers (to be defined more precisely later). C, The set of all complex numbers (also to be defined more precisely later). ∅ , the set with no elements, the empty or null set. It is extremely uninteresting. Definition: Given the two sets S and T , i) the set S ∪ T , “ S union T ”, is the set of elements which are in either S or T , or both. ii) The set S ∩ T , “ S intersection T ”, is the set of elements in both S and T. If we represent S by one blob and T by another, S ∪ T is the shaded region while S ∩ T is the cross-hatched region. Note that all elements in S ∩ T are also in S ∪ T. Two sets are disjoint if S ∩ T = ∅ , that is, if their intersection is empty.

1

2 CHAPTER 0. REMEMBRANCE OF THINGS PAST.

A subset of a set is another way of referring to a portion of a given set. Formally, A is the subset of S , written A ⊂ S , if every element in A is also an element of S. The set A is a subset of the set S if and only if either

A ∪ S = S, or, equivalently, A ∩ S = A.

It is possible that A = S , or that A = ∅. If these degenerate cases are excluded, we say that A is a proper subset of S. Given the two sets S and T , it is natural to form a new set S × T , “ S cross T ”, which consists of all pairs of elements, one from S and the other from T. For example, if S is the set of all men in this class, and T the set of all women in this class, then S × T is the set of all couples, a natural set to contemplate. If x ∈ S and y ∈ T , the standard notation for the induced element in S × T is (x, y). Note that the order in (x, y) is important. The element on the left is from S , while that on the right is from T. For this reason the pair of elements (x, y) is usually called an ordered pair. The whole set S × T is called the product, direct product, or Cartesian product of S and T , all three names being used interchangeably. You have met this idea in graphing points in the plane. Since these points, (x, y) , are determined by an ordered pair of real numbers, they are just the elements of R × R. From this example it is clear that even though this set R × R is the product of a set with itself, the order of the pair (x, y) is still important. For example the point (1, 2) ∈ R × R is certainly not the same as (2, 1) ∈ R × R. Having defined the direct product of two sets S and T as ordered pairs, it is reasonable to define the direct product of three sets S, T, and U as the set of ordered triplets (x, y, z) , where x ∈ S, y ∈ T , and z ∈ U. The extension to n sets, S 1 × S 2 × · · · × Sn , is done in the same way. Let us now recall the ideas behind the notion of a function. A function f from the set X into the set B is a rule which assigns to every x ∈ X one and only one element y = f (x) ∈ B. We shall also say that f maps X into B , and write either f : X → B, or X →f B.

This alternative notation is useful when X and B are more important than the specific nature of f. The set X is the domain of f , while the range of f is the subset Y ⊂ B of all elements y ∈ B which are the image of (at least) one point x ∈ X , so y = f (x) , or in suggestive notation, Y = f (X). Automobile license plates supply a nice example, for they assign to every license plate sold a unique car. The domain is the set of all license plates sold, while the range is not all cars, but rather the subset of all cars which are driven. Wrecks and museum pieces neither need nor have license plates since they are not on the roads. Some other examples are i) the function f (n) ≡ (^) n^1 , n = 1, 2 , 3 ,... which assigns to every n ∈ Z+ the rational number 1 n , and ii) the function^ f^ (n, m) =^

m n , n, m^ = 1,^2 ,^2 ,...^ , which assigns to every element of Z+ × Z+ the rational number mn. Quite often we shall use functions which map part of some set into part of some other set. In other words the function may be defined on only a subset of a given set and take on values in a subset of some other set. The function f (n, m) ≡ mn of the previous paragraph is of this nature for we defined it on a subset of Z × Z and takes its values on the positive subset of the set of all rational numbers.

4 CHAPTER 0. REMEMBRANCE OF THINGS PAST.

∃ “there is”, or “there exists” 3 “such that” A ⇒ B “the truth of statement A implies that of statement B ”. A ⇔ B “statement A is equivalent to statement B , that is, both A ⇒ B and B ⇒ A.

Exercises

(1) If R = { 1 , 4 }, S = { 1 , 2 , 3 , 4 , } , and T = { 2 , 3 , 7 } , find the six other sets R∪S, R∩ S, R ∪ T, R ∩ T, S ∪ T, and S ∩ T. Which of these nine sets are proper subsets of which other sets?

(2) If S = { x : |x − 1 | ≤ 2 } and T = { x : |x| ≤ 2 } , find S ∪ T and S ∩ T. A sketch is adequate.

(3) If A, B , and C are any subsets of a set S , prove

(a) (A ∪ B) ∪ C = A ∪ (B ∪ C) —so that the parenthesis can be omitted without creating ambiguity. (b) (A ∩ B) ∩ C = A ∩ (B ∩ C) —so that again the parentheses are superfluous. (c) (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C). (d) (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C).

Remark: two sets X and Y are proved equal by showing that both X ⊂ Y and Y ⊂ X.

(4) If the function f has domain S , and both A ⊂ C and B ⊂ S , prove that

(i) A ⊂ B ⇒ f (A) ⊂ f (B). (ii) f (A ∩ B) ⊂ f (A) ∩ f (B) [We cannot hope to prove equality because of coun- terexamples like: let A = { − 2 , − 1 , 0 , 1 , 2 , 3 } and B = { − 4 , − 3 , − 2 , − 1 }. Then with f (n) = n^2 , we have f (A) = { 0 , 1 , 4 , 9 }, f (B) = { 1 , 4 , 9 , 16 } , and f (A ∪ B) = { 1 , 4 } 6 = f (A) ∩ f (B) ]. (iii) f (A ∪ B) = f (A) ∪ f (B).

(5) For the following functions f : X → B , classify as to injection, surjection, or bijection, or none of these.

(i) f (n) = n^2 with X = Z+ and B = Z. (ii) Let X = { all rational numbers } , B = { all rational numbers } , and f (x) = (^) m^1 , where x = (^) mn ∈ X [Here (^) mn is assumed to be reduced to lowest terms.] (iii) f (x) = (^) x^1 , where x ∈ X and X = B = { all positive rational numbers }. (iv) X = { all women born in May }, B = { the thirty days in the month of June } , and let f be the function assigning “her birthday” to each woman born in June. (v) f (n) = |n| , with X = B = Z.

0.2. RELATIONS 5

0.2 Relations

A relationship often exists between elements of sets. Some common examples are i) a ≥ b , ii) a ⊥ b (perpendicular to), iii) a loves b , and iv) a 6 = b. Let S be a given set, a , b ∈ S , and let R be a relation defined on S (that is, ∀ a, b ∈ S , either aRb or aRb with no third alternative possible). Most relations have at least one of the following properties.

(i) reflexive aRa ∀a ∈ S

(ii) symmetric aRb ⇒ bRa

(iii) transitive (aRb and bRc) ⇒ aRc.

Examples:

(1) perpendicular ( ⊥ ) is only symmetric.

(2) “loves” enjoys none of these (well, maybe it is reflexive).

(3) equality ( = ) has all three properties.

(4) geometric congruence (∼=) and geometric similarity (') both have all three.

(5) parallel (‖) has all three—if we are willing to agree that a line is parallel to itself.

(6) “is less than five miles from” is only reflexive and symmetric.

(7) for a, b ∈ Z+ , the relation “ a is divisible by b ” is only reflexive and transitive but not symmetric.

(8) “less than” ( < ) is only transitive.

A relation which is reflexive, symmetric and transitive is called an equivalence relation. The standard examples are those of algebraic equality and of geometric congruence. An equivalence relation on a set S partitions the set into subsets of equivalent elements. Those terms are illustrated in the following.

Examples:

(1) In the set S of all triangles, the equivalence relation of geometric congruence parti- tions S into subsets of congruent triangles, any two triangles of S being in the same subset (or equivalence class as it is called) if and only i f they are congruent.

(2) In the set P of all people, consider the equivalence relation ”has the same birthday,” disregarding the year. This relation partitions P into 366 equivalence classes. Two people are in the same equivalence class if their birthdays fall on the same day of the year.

Notice that any two equivalence classes are either identical or disjoint, that is, they have either no elements in common or they coincide. This is particularly clear from the examples with birthdays. By the fundamental theorem of calculus, we know that the indefinite integral of an integrable function f can be represented by any function F whose derivative is f. The

0.4. REALS: ALGEBRAIC AND ORDER PROPERTIES 7

The formula, assumed to be true, for n = k is

1 + 2 + · · · + k = k(k + 1) 2

Adding (k + 1) to both sides we find that

1 + 2 + · · · + k + (k + 1) = k(k + 1) 2

• (k + 1) = (k + 1)(k + 2) 2

which is exactly the statement we wanted. This proves that formula (0.3) is true for all n ≥ 1.

Exercises

Use mathematical induction to prove the given statements.

(1) 12 + 2^2 + · · · + n^2 = n(n+1)(2 6 n+1)

(2) (^) dxd (xn) = nxn−^1 (use the formula for the derivative of a product).

(3) Let I(n) =

∫ π 2 0 sin

n (^) x dx

(a) Prove the following formula is correct when n is an odd integer ≥ 3 ,

I(n) =

2 · 4 · 6 · · · ·(n − 1) 1 · 3 · 5 · · · ·n

(b) Guess and prove the formula when n is an even integer ≥ 2.

(4) Let Γ(s) =

0 e

−tts− (^1) dt , where s > 0 (this is the famous gamma function).

(a) Show Γ(s + 1) = sΓ(s) (Hint: integrate by parts) (b) If n ∈ Z+ , guess and prove the formula for Γ(n + 1).

0.4 The Real Numbers: Algebraic and Order Properties.

The set of all real numbers can be characterized by a set of axioms. These properties are of three different types, i) algebraic properties, ii) order properties, and iii) the completeness property. Of these, the last is by far the most difficult to grasp. But that is getting ahead of our story. Let S be a set with the following properties. I. Algebraic Properties A. Addition. To every pair of elements a, b ∈ S , is associated another element, denoted by a + b , with the properties A - 0. (a + b) ∈ S A - 1. Associative: for every a, b, c ∈ S, a + (b + c) = (a + b) + c. A - 2. Commutative: a + b = b + a A - 3. There is an additive identity, that is, an element ”0” ∈ S such that 0 + a = a for all a ∈ S. A - 4. For every a ∈ S , there is also a b ∈ S such that a + b = 0. b is the additive inverse of a , usually written −a.

8 CHAPTER 0. REMEMBRANCE OF THINGS PAST.

M. Multiplication. To every pair a, b ∈ S , there is associated another element, denoted by ab , with the properties M - 0. ab ∈ S M - 1. Associative. For every a, b, c ∈ S, a(bc) = (ab)c. M - 2. Commutative. ab = ba. M - 3. There is a multiplicative identity, that is, an element “ l” ∈ S such that la = a for all a ∈ S. Moreover 1 6 = 0. M - 4. For every a ∈ S, a 6 = 0 , there is also a b ∈ S such that ab = 1. b is the multiplicative inverse of a , usually written (^1) a or a−^1. D. Connection between Addition and Multiplication. D - 1. Distributive. For every a, b, c ∈ S, a(b + c) = ab + ac. Some sample - and simple—consequences of these nine axioms are i) a + 0 = a , ii) a · 1 = a , and iii) a + b = a + c ⇒ b = c. Any set whose elements satisfy the axioms A-0 to A-4 is called a commutative(or abelian) group. The group operation here is addition. In this language, we see that the multiplication axioms just state that the elements of S —with the additive identity 0 excluded—also form a commutative group, with the group operation being multiplication. These additive and multiplicative structures are connected by the distributive axiom. Most of high school algebra takes place in this setting; however, the possibility of non-integer exponents is not yet specifically included; in particular the square root of an element of S is not necessarily also in S. Our axioms, or some part of them, are satisfied by sets other than the real numbers. The set of even integers form a commutative group with the group operation being addition, while numbers of the form 2 n, n ∈ Z , form a commutative group under multiplication. The set of rational numbers satisfies all nine axioms. Any such set which satisfies all nine axioms is called a field. Both the real numbers and the rational numbers (a subset of the real numbers) are fields. A more thorough investigation of groups and fields is carried out in courses in modern algebra. II. Order Axioms Besides the above algebraic rules, we shall introduce an order relation, intuitively, the notion of ’greater than”. To do this we need to use an undefined concept of positivity for elements of S and use it to state our axioms. O -1. If a ∈ S and b ∈ S are positive, so are a + b and ab. O -2. The additive identity 0 is not positive. O - 3. For every a ∈ S, a 6 = 0 , either a or −a is positive, but not both. If −a is positive, we shall say that a is negative.

Trichotomy Theorem. For any two numbers a, b ∈ S , exactly one of the following three statements is true, i) a − b is positive, ii) b − a is positive, or iii) b − a is zero. If the notation a < b is used to mean “ b − a is positive,” and a > b means b < a , then this theorem reads, either a > b, a < b, or a = b. The proof—which you should do—is a simple consequence of our axioms. Some other consequences are a < b and b < c ⇒ a < c (transitivity of “ < ”) a < b and c > 0 ⇒ ac < bc a 6 = 0 ⇒ a^2 > 0. (Since 1 = 1^2 , this implies 1 > 0 ). The set of rational numbers as well as the set R of real numbers satisfy all twelve axioms. Any set which satisfies these twelve axioms is called an ordered field.

10 CHAPTER 0. REMEMBRANCE OF THINGS PAST.

Cauchy sequence version of the completeness axiom, we would have begun the rational numbers—which we do know—and then defined the real numbers as the set of limits of rational numbers. This would have been somewhat more concrete, but would have involved the difficult concept of limit before we even get off the ground. From the picture associated with the completeness axiom, we see that it exactly states that the real number line has no holes, for - emotionally speaking—if there were a hole, let S 1 be the set of real numbers to the left of the hole, and S 2 the s et to the right of the hole. Then there would be no real number between S 1 and S 2 , since the hole is there, contradicting the completeness axiom. Let us use the idea of the last paragraph to show that the rational numbers, an ordered field, are not complete by exhibiting two sets, one preceding the other, which have no rational number between them. Just let

S 1 = { x : x > 0 , x^2 < 2 } and S 2 = { x : x > 0 , x^2 > 2 }.

The only possible number between S 1 and S 2 is

2 —which is irrational. This construc- tion is just what we need to prove the following sample.

Theorem 0.1 Every non-negative real number a ∈ R has a unique non-negative square root.

Proof: If a = 0 , then 0 is the square root. If a > 0 , let S 1 = { x : x > 0 , x^2 < a } and S 2 = { x : x > 0 , x^2 > a }. We first show that neither S 1 nor S 2 is empty. Since (1 + a 2 )^2 = 1 + a + a 2 4 > a^ , we know that (1 +^

a 2 )^ ∈^ S^2 , so^ S^2 6 =^ ∅^. Also (^

a 1+ a 2 )

(^2) < a (check

this) so that (^) 1+a a 2 ∈ S 1 and hence S 1 6 = ∅. Because S 1 precedes S 2 , by the completeness

axiom there is a c ∈ R between S 1 and S 2. Notice that c > 0 , since c is preceded by S 1. It remains to show that c^2 = a. By the trichotomy theorem, either c^2 > a, c^2 < a , or c^2 = a. The first two possibilities will be shown to give contradictions. If c^2 > a , since a < ( c (^2) +a 2 c )

(^2) < c (^2) , we see that c^2 +a 2 c ∈^ S^2 an d precedes^ c

(^2) , contradicting the property

specified in the completeness axiom that c^2 precedes every element of S 2. Similarly the assumption c^2 < a , with the inequality c^2 < ( (^) c^22 ac+a )^2 < a , leads to a contradiction. The only remaining possibility is c^2 = a , which shows that c is the desired positive square root of a. Let us now prove that the positive square root c of a is unique. Assume that there are two positive numbers c 1 and c 2 such that both c^21 = a and c^22 = a. Then

0 = c^21 − c^22 = (c 1 − c 2 )(c 1 + c 2 )

Since c 1 + c 2 > 0 , we conclude that c 1 − c 2 = 0 , so c 1 = c 2 , completing the proof of the theorem. Definition: The real number M is an upper bound for the set A ⊂ R if for every a ∈ A , we have a ≤ M. The number μ ⊂ R is a least upper bound (l.u.b) for A if μ is an upper bound for A and no smaller number is also an upper bound for A. Lower bound and greatest lower bound (g.l.b) are defined similarly. A set A ⊂ R is bounded if it has both upper and lower bounds.

Theorem 0.2 Every non-empty bounded set A ⊂ R has both a greatest lower bound and a least upper bound.

0.5. REALS: COMPLETENESS 11

Proof: Observe first that this theorem utilizes the completeness property in that without it, there might have been a ”hole” just where the g.l.b. and l.u.b. should be. Since the proofs for the g.l.b. and l.u.b. are almost identical we only prove there is a g.l.b. Let

S 1 = { x : x precedes A }, and S 2 = A.

By hypothesis S 2 6 = ∅. Since A is bounded, it has a lower bound m, m ∈ S 1 so S 1 6 = ∅. By the completeness axiom, there is a c ∈ R between S 1 and S 2. It should be obvious that c is both greater than or equal to every element of S 1 , and less than or equal to every element of S 2 - so it is the required g.l.b. Definition: The closed interval [a,b] is the set { x ∈ R : a ≤ x ≤ }. The open interval (a,b) is the set { x ∈ R : a < x < }. All we can do is apologize for the multiple use of the parentheses in notation. Please note that sets are not like doors. Some sets, like (a, b) = { x ∈ R : a ≤ x < } are neither open nor closed.

Theorem 0.3 (Nested set property). Let I 1 , I 2 ,... be a sequence of non-empty closed bounded intervals, In = { x : an ≤ x ≤ bn } , which are nested in the sense I 1 ⊃ I 2 ⊃ I 3... , so each covers all that follow it. Then there is at least one point c ∈ R which lies in all of the intervals, that is, c is in their intersection c ∈ ∩∞ k=1Ik.

Proof: Let S 1 = { x : x precedes some In, and so all Ik, k ≥ n } S 2 = { x : x preceded by some In, and so all Ik, k ≥ n }. First, neither S 1 nor S 2 are empty since a 1 ∈ S 1 and b 1 ∈ S 2. Thus by the completeness axiom, there is at least one c ∈ R between S 1 and S 2. This c is the required number (complete the reasoning). If the intervals Ik do not get smaller after, say IN because aN = aN +1 =... and bN = bN +1 =... , then the whole interval aN ≤ x ≤ bN is caught by the preceding argument. The more common case is there the ak ’s strictly increase and the bk ’s strictly decrease. This is what happens when approximating a real number to successively greater accuracy by the decimal expansion. In the case of

2 for example, I 1 = { x : 1 ≤ x ≤ 2 }, I 2 = { x : 1. 4 ≤ x ≤ 1. 5 }, I 3 = { x : 1. 41 ≤ x ≤ 1. 42 }, I 4 = { x : 1. 414 ≤ x ≤ 1. 415 }, and so on, gradually squeezing down on

2 to any desired accuracy. Definition: The sequence an ∈ R, n = 1 , 2 ,.... of real numbers converges to the real number c if, given any  > 0 , there is an integer N such that |an − c| <  for all n > N. We will then write an → c. [In practice no confusion arises for the use of → to denote both convergence and mappings (cf. 1)]. Again ordinary decimals supply an example, for they allow us to get arbitrarily close to any real number. We could have defined the real numbers as all decimals; however there would be a mess avoiding the built-in ambiguity illustrated by 1. 9999.... = 2. 0000....

Theorem 0.4 Under the hypotheses of the previous theorem, if in addition the length of In tends to zero, (bn − an) → 0 , then the number c ∈ R found is unique. Furthermore, if uk ∈ Ik for all k , that is if ak ≤ uk ≤ bk , then uk → c too.

Proof: Suppose there were two real numbers c and ˜c in all of the intervals,

ak ≤ c ≤ bk and ak ≤ ˜c ≤ bk for all k.