IntroRealAnalysNotes.pdf

Notes in Introductory Real Analysis Ambar N. Sengupta March, 2014 2 Ambar N. Sengupta Contents Introductory Remarks........................... 5 1 Ordered Fields and The Real Number System...

Notes in Introductory Real Analysis Ambar N. Sengupta March, 2014 2 Ambar N. Sengupta Contents Introductory Remarks........................... 5 1 Ordered Fields and The Real Number System 7 1.1 Ordered Fields........................... 8 1.1.1 Fields............................ 9 1.1.2 Order Relations....................... 13 1.1.3 Ordered Fields....................... 14 1.1.4 The absolute value function................ 16 1.1.5 The Archimedean Property................. 18 1.2 The Real Number System R.................... 18 1.2.1 Hilbert Maximality and the Completeness Property.... 19 1.2.2 Completeness of R and measurement........... 20 Problem Set 1............................ 20 2 The Extended Real Line and Its Topology 23 2.1 The extended real line........................ 23 2.2 Neighborhoods........................... 24 2.3 Types of points for a set....................... 25 2.4 Interior, Exterior, and Boundary of a Set.............. 27 2.5 Open Sets and Topology...................... 28 2.6 Closed Sets............................. 30 2.7 Open Sets and Closed Sets..................... 30 2.8 Closed sets in R and in R∗..................... 31 2.9 Closure of a set........................... 31 2.10 The closure of a set is closed.................... 32 2.11 S is the smallest closed set containing S.............. 32 2.12 R∗ is compact............................ 32 2.13 Compactness of closed subsets of R∗................ 35 2.14 The Heine-Borel Theorem: Compact subsets of R......... 36 3 4 Ambar N. Sengupta 2.15 Sequences.............................. 37 2.16 Limits points of a sequence..................... 38 2.17 Bolzano-Weierstrass Theorem................... 39 2.18 Limit points and suprema and infima................ 39 2.19 Limit of a sequence......................... 41 2.20 Simple Examples of limits..................... 42 2.21 The sequence 1/n.......................... 43 2.22 The sequence Rn.......................... 44 2.23 Monotone Sequences........................ 48 2.24 The limit of a sequence is unique.................. 49 2.25 Convergent sequences and Cauchy sequences........... 49 2.26 Every Cauchy sequence is bounded................. 50 2.27 Every Cauchy sequence is convergent............... 51 2.28 The rationals are countable..................... 52 2.29 The real numbers are uncountable................. 54 2.30 Connected Sets........................... 55 3 Integration 57 3.1 Approaching the Riemann Integral................. 58 3.2 Riemann Sums........................... 59 3.3 Definition of the Riemann Integral................. 61 3.4 Refining Partitions......................... 62 3.5 The Darboux Criterion....................... 65 3.6 Integrable functions are bounded.................. 66 3.7 Variation of a Function....................... 67 3.8 The algebra R [a, b]......................... 70 3.9 C[a, b] ⊂ R [a, b]........................... 72 3.10 The Integral as a Non-negative Linear Functional......... 73 3.11 Additivity of the Integral...................... 79 3.12 Monotone Functions are Riemann Integrable............ 81 3.13 Riemann Sums and the Riemann Integral............. 83 Bibliography............................ 86 A Question Bank 89 Notes in Introductory Real Analysis 5 Introductory Remarks These notes were written for an introductory real analysis class, Math 4031, at LSU in the Fall of 2006. In addition to these notes, a set of notes by Professor L. Richardson were used. There are several different ideologies that would guide the presentation of concepts and proofs in any course in real analysis: (i) the historical way (ii) the most natural way (iii) the most efficient way (iv) a comprehensive way, explaining the insights from several different ap- proaches The reality of constraints of time makes (iii) the most convenient approach, and perhaps the best example of this approach is Rudin’s Principles of Mathemat- ical Analysis. The downside is that there is little possibility of conveying any insights or intuition. In studying the notion of completeness, a choice has to be made whether to treat the Cauchy sequence point of view or the existence of suprema as funda- mental. I have chosen the latter; it conforms to the classical geometric notion of a positive real number being specified by quantities greater than it and those less than it. This point of view also guides the choice of approach in the treatment of the Riemann integral; the Riemann integral of a function is the unique real number lying between the upper Riemann sums and lower Riemann sums. The notes here do not include a chapter on continuous functions, for which we followed the Richardson notes. These notes have not been proof read carefully. I will update them time to time. Comments from many students have helped improve the notes. Among those who deserve thanks are John Tate (in-class comments) and Daniel Donovan (email 2014). 6 Ambar N. Sengupta Chapter 1 Ordered Fields and The Real Number System In this chapter we go over the essential, foundational, facts about the real number system. Positive real numbers arose from geometry in Greek mathematics, as ratios of magnitudes, such as segments or planar regions or even angles. In the discussion below we focus on segments. In Euclid’s Elements, a segment EF is taken to exceed a segment GH, symbol- ically EF>GH if EF is congruent to a segment GK, where K is some point between E and F. An important feature of this order relation is encapsulated in the archimedean axiom: given any two segments, some multiple of any one of them exceeds the other. Then aAa pair PQ and RS if for any positive numbers n and m, the segment nAB (which is n copies of AB laid contiguously) exceeds the segment mCD if and only if the segment nPQ exceeds the segment mRS. The ratio AB CD is then essentially the equivalence class of all segment pairs which are in the same ratio as AB is to CD. Euclid also defines the ratio XY/ZW to be greater than the ratio PQ/RS : XY PQ > ZW RS if they are unequal and if whenever mZW>nXY then also mRS>nPQ. 7 8 Ambar N. Sengupta A special case is that of commensurate segments: if a whole multiple of AB, say nAB, is congruent to a whole multiple of CD, say mCD, then the ratio ABCD is rational, and is denoted by m. n It is readily checked that m p = n q if and only if qm = pn. Such ratios are the rational numbers. Other ratios are irrational. In either case, Euclid’s considerations suggest that a ratio of segments may be understood in terms of a set of rational numbers, for example the set of all those rationals which exceed the given ratio. The axioms of geometry, and the Euclidean construction procedures, show that ratios of segments can be added and multiplied and, when 0 and negatives are included, an algebraic structure called a field emerges (this is discussed at length by Hilbert ). The maximal such field, respecting the axioms of geometry pertaining to the order relation and congruence, constitutes the real number system R. Leaving aside the historical background, the real number system may be con- structed by starting with the empty set, constructing the natural numbers, then the rationals, and then the real numbers by Dedekind’s method of identifying a real number with a splitting of the rationals into two disjoint classes with members of one class exceeding those of the other. Dedekind’s method has beed generalized in a striking, and vastly more power- ful way, by Conway , who shows how the Dedekind cut method can be applied to abstract sets leading to the construction of all real numbers as well as tran- scendentals and infinitesimals. Knuth’s novel is an unusual and entertaining presentation of this construction. 1.1 Ordered Fields In this section we define and prove simple properties of fields, ordered fields, and absolute values. The reader wishing to move on to properties of the real numbers may skim the contents of the first few subsections, and proceed to subsection 1.1.4. Notes in Introductory Real Analysis 9 1.1.1 Fields A field F is a set along with two binary operations Addition : F × F → F : (a, b) 7→ a + b (1.1) and Multiplication : F × F → F : (a, b) 7→ ab (1.2) satisfying the following axioms: 1. The associative law holds for addition a + (b + c) = (a + b) + c for all a, b, c ∈ F (1.3) 2. There is an element 0 ∈ F, the zero or additive identity element, for which a+0 = a = 0+a for all a ∈ F (1.4) 3. Every element a ∈ F has an additive inverse −a, called the negative of a: a + (−a) = 0 = (−a) + a (1.5) 4. The commutative law holds for addition: a+b = b+a for all a, b ∈ F (1.6) 5. The associative law holds for multiplication 6. The associative law holds for addition a(bc) = (ab)c for all a, b, c ∈ F (1.7) 7. There is an element 1 ∈ F, the unit or multiplicative identity element, for which a1 = a = 1a for all a ∈ F (1.8) 8. Every non-zero element a ∈ F has an multiplicative inverse a−1 , called the reciprocal of a: aa−1 = 1 = a−1 a for all non-zero a ∈ F (1.9) 10 Ambar N. Sengupta 9. The commutative law holds for multiplication: ab = ba for all a, b ∈ F (1.10) 10. The distributive law holds: a(b + c) = ab + ac, (b + c)a = ba + ca for all a, b, c ∈ F (1.11) 11. The element 1 is not equal to the element 0: 1 6= 0 We have not attempted to provide a minimal axiom set, and some of the axioms may be deduced from others. For instance, the commutativity of addition can be deduced from the other axioms. Because of the associative laws, we will just write a+b+c instead of a + (b + c), and abc instead of abc. Let us note a few simple consequences: Theorem 1 If x ∈ F is such that a+x = a for some a ∈ F and y ∈ F is such that by = b for some non-zero b ∈ F then x=0 and y = 1. In particular, the additive identity and the multiplicative identity are unique. More- over, −0 = 0 and 1−1 = 1 Notes in Introductory Real Analysis 11 Proof. Adding −a to a+x = a shows that x is 0. Multiplying by = b by b−1 shows that y is 1. The other claims follow from 0+0 = 0 and 1 · 1 = 1. QED Theorem 2 If a, b ∈ F, and b 6= 0, then −(−a) = a, and (b−1 )−1 = b. Moreover, (−a)b = −ab, and (−a)(−b) = ab. The multiplicative inverse a−1 is best written as the reciprocal: 1 = b−1 , b and the product ab−1 as a = ab−1 b There are many other easy consequences of the axioms which we will use without comment. We denote the set of natural numbers by P: P = {1, 2, 3,...}, (1.12) and the set of integers by Z = {0, 1, −1, 2, −2, 3, −3,...}, (1.13) We can multiply any element a ∈ F by an integer as follows. First define 1a = a, where now 1 is the number one in Z. Next, def 2a = a + a, 12 Ambar N. Sengupta and, inductively, def (n + 1)a = na + a (1.14) for all a ∈ F and all n ∈ P. Next define negative multiples by (−n)a = −(na) (1.15) for all n ∈ P and all a ∈ F. Finally, 0a = a (1.16) where 0 is the integer 0. The following facts can be readily verified: x(ya) = (xy)a for all x, y ∈ Z and all a ∈ F (1.17) x(a + b) = xa + xb for all x ∈ Z and all a, b ∈ F (1.18) (x + y)a = xa + ya for all x, y ∈ Z and all a ∈ F (1.19) There is a multiplicative analog of this given by powers of elements. If m is a positive integer and a ∈ F we define a1 = a and am+1 = am a. We also define a0 = 1 for non-zero a ∈ F and, for any positive integer m and non-zero a ∈ F, 1 a−m = (a−1 )n = am We will use without comment simple facts such as (am )n = amn , valid for suitable a ∈ F and m, n ∈ Z. The simplest example of a field is the two-element field Z2 = {0, 1}, with addition and multiplication defined modulo 2; for example, 1+1 = 0 in Z2 We will, however, be concerned with fields which permit a consistent ordering of their elements. Notes in Introductory Real Analysis 13 1.1.2 Order Relations An order relation on a set S is a set O of ordered pairs (x, y) of elements of S sat- isfying the conditions O1 and O2 below. It is convenient to adopt the convention that x < y means (x, y) ∈ O We also write y>x to mean x < y. The axioms of order are: O1. For any x, y ∈ F exactly one of the following hold: x = y, or x < y, or y < x. O2. If x < y and y < z then x < z. It is also convenient to use the notation: x ≤ y means x = y or x < y and, similarly, x ≥ y means x = y or x > y If T is a subset of an ordered set S then an element u ∈ S is said to be an upper bound of T if t ≤ u for all t ∈ T (1.20) If there is a least such upper bound then that element is called the supremum of T : sup T = the least upper bound of T (1.21) We define similarly lower bounds and infimums: if l ≤ t for every t ∈ T then l is called a lower bound of T (1.22) and inf T = the greatest lower bound of T (1.23) Of course, the sup or the inf might not exist. 14 Ambar N. Sengupta 1.1.3 Ordered Fields An ordered field is a field F with an order relation in which, in addition to the field and order axioms stated above, the following hold: OF1. If x, y, z ∈ F and x < y then x + z < y + z: x < y implies x + z < y + z for all x, y, z ∈ F (1.24) OF2. If x, y, z ∈ F and x < y, and if also z > 0, then xz < yz: x < y and z > 0 imply xz < yz for all x, y, z ∈ F. (1.25) If x > 0 we say that x is positive. If x < 0 we say that x is negative. We have the following simple observations for an ordered field: Theorem 3 Let F be an ordered field. Then (i) x > 0 if and only if −x < 0 (ii) For any non-zero x ∈ F we have x2 > 0 (iii) 1 > 0 (iv) For any r ∈ Z the element r1 ∈ F is > 0 if r is a positive integer and is < 0 if r is a negative integer (v) x > y holds if and only if x − y > 0 (vi) If x ∈ F and x > 0 then 1/x > 0 (vii) The product of two positive elements is positive (viii) The product of a positive and negative is negative (ix) The product of two negative elements is positive (x) If x > y and z < 0 then xz < yz. (xi) If x > y then −x < −y (xii) If x > y > 0 then 1/x < 1/y Notes in Introductory Real Analysis 15 Proof. We prove some of the results. (i) Suppose x > 0. Then we have x + (−x) > 0 + (−x), and so 0 > −x. Conversely, if −x < 0 then adding x to both sides shows that x > 0. (ii) If x > 0 then, by OF2, we have x2 = xx > x0 = 0. On the other hand, if x < 0 then we know that −x > 0 and so x2 = (−x)(−x) > (−x)0 = 0. (iii) Since 1 is 12 , it follows that 1 > 0. (vi) Suppose x > 0. Since x(1/x) = 1, and 1 6= 0, it follows that 1/x cannot be zero. If 1/x < 0 then, however, x(1/x) would have to be < 0, but we know that 1 > 0. Thus, 1/x > 0. (xi) If x > y then, adding −x − y shows that −y < −x. (xii) If x > y > 0 then xy > 0 and hence 1/(xy) > 0. Multiplying x > y by the positive element 1/(xy) gives 1/y > 1/x. QED Observe that if x > 0 then 2x = x + x > x + 0 = x, and 3x = 2x + x > 2x + 0 = 2x, and, proceeding inductively, we have 0 < x < 2x < 3x < · · · < nx < (n + 1)x < · · · for all n ∈ P and x > 0 in F (1.26) In particular, inside the ordered field F we have a copy of the natural numbers 1, 2, 3,... on identifying n ∈ P with n1F , where 1F is the unit element in F. This then leads to a copy of the integers inside F, and we can assume that Z⊂F (1.27) 16 Ambar N. Sengupta Going further, if m, n ∈ Z, and n 6= 0, we have then the ratio m ∈ F. n The set of all such ratios m/n is the set of rationals Q⊂F (1.28) and is an ordered subfield of the ordered field F. 1.1.4 The absolute value function The absolute value | · | function in an ordered field F is defined by ( x if x ≥ 0 |x| = (1.29) −x if x ≤ 0 For instance, |1| = 1, and | − 1| = 1. The definition of |x| shows directly that | − x| = |x| ≥ 0 for all x ∈ F (1.30) It is also useful to observe that |x| is the larger of x and −x (1.31) We think of |a − b| as measuring the difference between a and b. We have then Theorem 4 For any a, b ∈ F we have: (i) the triangle inequality |a + b| ≤ |a| + |b| (1.32) Equality holds if and only if a and b are both ≥ 0 or both ≤ 0. Notes in Introductory Real Analysis 17 (ii) the absolute values differ by at most the difference in a and b: |a| − |b| ≤ |a − b| (1.33) (iii) |ab| = |a||b|. Proof. Recall that |x| is the larger of x and −x. Therefore, |a| + |b| is greater or equal to ±a + (±b); in particular, it is greater or equal to a + b and (−a) + (−b). Consequently, |a| + |b| is greater or equal to a + b and −(a + b), and so |a| + |b| ≥ |a + b|. Equality holds if and only if |a| = a and |b| = b or |a| = −a and |b| = −b. Thus, equality holds if and only if a and b are either both ≥ 0 or both ≤ 0. Next, using the triangle inequality we have |a − b| + |b| ≥ |a + b − b| = |a|, and so |a − b| ≥ |a| − |b|. Interchanging a and b yields: |b − a| ≥ |b| − |a|. Observe that |b − a| = |a − b|. Thus, |a − b| is ≥ to both |a| − |b| and −(|a| − |b|). Since the larger of the latter is |a| − |b| , we have proved (1.33). We know that |x| is x or −x, whichever is ≥ 0. Consequently, the product |a||b| is one of the elements ab, (−a)b, a(−b), (−a)(−b), i.e. one of the elements ab and −ab. Thus, |a||b| is ab or −ab, whichever is ≥ 0, and so it is |ab|. QED 18 Ambar N. Sengupta 1.1.5 The Archimedean Property An ordered field F is said to have the archimedean property if for any x, y ∈ F, with x > 0, there exists a multiple of x which exceeds y: nx > y for some n ∈ {1, 2, 3,...} (1.34) The field Q of rationals is clearly archimedean: Theorem 5 The ordered field Q of rationals is acrhimedean. Proof. Let x, y ∈ Q, with x > 0. If y ≥ 0 then we have 1x > y and we are done. Now suppose x, y > 0. Then p r x= y= , q s where p, q, r, s ∈ P. Take n = qr + 1. Then r nx = qr(p/q) + p/q > pr ≥ = y, s and we are done. QED In an archimedean ordered field there are no infinities, and there are also no infinitesimals: Theorem 6 If F is an archimedean field then for any w > 0 in F there is an n ∈ P such that 1 w < x. n Proof. Simply choose n for which nx is > w. QED 1.2 The Real Number System R We shall work with the real number system in an axiomatic way. We will assume that it is an ordered field in which the completeness axiom of completeness holds. Needless to say, it is essential to actually construct such a system so as to be sure that there is no hidden contradiction between the axioms, but we shall not describe a construction in these notes. Notes in Introductory Real Analysis 19 1.2.1 Hilbert Maximality and the Completeness Property As we have mentioned before, the structure of Euclidean geometry, as formalized through the axioms of Hilbert, produces an archimedean ordered field. To com- plete the story, one can add to these axioms the further requirement that this field is maximal in the sense that it cannot be embedded inside any larger archimedean ordered field. It turns out then that any such ordered field is isomorphic to any other, and thus there is essentially one such ordered field. This ordered field is the real number system R. A crucial fact about R is the completeness property: If L and U are non-empty subsets of R such that every element of L is ≤ every element of U, then there is a real number m which lies between L and U: l ≤ m ≤ u for all l ∈ L and all u ∈ U. (1.35) This property is also often expressed as: If R is partitioned into two disjoint subsets L and U whose union is R, and if every element of L is ≤ every element of U then there is a unique element x ∈ R which lies between L and U: l ≤ x ≤ u for all l ∈ L and all u ∈ U. (1.36) It is useful to view the real numbers geometrically. Consider a line, with two special points O and I marked on it. For any point P on the line on the same side of O as P we think of the ratio OP/OI as a positive real number. Points on the other side from I correspond to negative real numbers, and the point O itself should be thought of as 0. The completebess property says that there are no ‘gaps’ in the line. The completeness property can be formulated equivalently as: Every non-empty subset of R which has an upper bound has a least upper bound. (1.37) The completeness property implies the archimedean peoperty: Theorem 7 If in an ordered field the property (1.37) holds then the archimedean property also holds. 20 Ambar N. Sengupta A proof of this is outlined in an exercise below. The modern understanding and point of view on completeness grew out of the work of Richard Dedekind. 1.2.2 Completeness of R and measurement Even in Euclid’s geometry, a real number was, implicitly, understood in terms of all rationals which exceeded it and all rationals below it. However, the traditional axioms of Euclidean geometry, with requirements on intersections of lines and circles, can work with a field which is larger than the rationals but smaller than R, and completeness is not essential. The simplest measurement problem is to devise a measure of sets of points in the plane which are made up of a finite collection of segments constructed by Euclidean geometry. Two such sets should have the same measure if they can be decomposed into a finite collection of congruent segments. For this we would not need the full complete system R of real numbers. Moving up a dimension, with the task of measuring areas of polygonal regions constructed by Euclidean geometry, one could still get away with a less-than-complete system of numbers. However, it was shown by Max Dehn in 1900, in resolving Hilbert’s Third Prob- lem, that there are polyedra in three dimensions which have equal volumes (as defined by requirements of ‘upper’ and ‘lower’ approximations) which cannot be decomposed into congruent pieces. This, along with, of course, the utility of measuring areas of curved regions even in two dimensions, makes it absolutely essential to work with a notion of measure that goes beyond simply decomposing into geometrically congruent figures. For a truly useful theory of measure, the completeness of the number system is essential. Capturing a real number between upper approximations and lower approxima- tions proves to be very useful. Archimedes and others computed areas of curved regions by such upper and lower approximations. In modern calculus, this method lives on in the Riemann integral, as we shall see later. Problem Set 1 1. Prove that in any ordered field, between any two distinct elements there is at least one other element. 2. Prove that in any ordered field, between any two distinct element lie in- finitely many elements. Notes in Introductory Real Analysis 21 3. If F is an archimedean ordered field, such as R, show that between any two distinct elements there is a rational element. 4. Let F be an archimedean ordered field, and let x, y ∈ F with x < y. If z ∈ F show that there is a rational multiple qz of z for which x < qz < y. 5. Suppose x ∈ R is ≥ all elements of a non-empty subset S ⊂ R. Explain why this implies x ≥ sup S. 6. Suppose S is a non-empty subset of R and x ∈ R is not an upper bound of S. Show that there exists y > x which is also not an upper bound of S. 7. Prove that the completeness property (1.37) implies the property (1.36). [Hint: Suppose (1.37) holds. If S ⊂ R is a non-empty set which is bounded above, then we can take U to be the set of all upper bounds of S and L to be the complement, i.e. all real numbers which are < some element of S. Let x ∈ F be provided by (1.36). Show that x is the least upper bound of S.] 8. Prove that (1.36) implies the completeness property (1.37). 9. Prove that the completeness property implies the archimdean propery. [Hint: Suppose F is an ordered field in which (1.37) holds. Let x, y ∈ F, with x > 0. Suppose that no positive integral multiple of x exceeds y, and let S = {nx : n ∈ P}. Then y is, by assumption, an upper bound for S. Let u = sup S, the least upper bound of S. Consider the element u − 21 x. This, being < u, is not an upper bound. Use this to produce an element of S greater than u, thus reaching a contradiction.] 10. Suppose L and U are non-empty subsets of R such that (i) every element of L is ≤ every element of R, and (ii) for any ε > 0 there is an element l ∈ L and an element u ∈ U with u − l < ε. Prove that there is a unique real number which is ≤ all elements of U and ≥ all elements of L. 22 Ambar N. Sengupta Chapter 2 The Extended Real Line and Its Topology In this chapter we study topological concepts in the context of the real line. For technical purposes, it will be convenient to extend the real line R by adjoining to it a largest element ∞ and a smallest element −∞. No metaphysical meaning need be attached to these infinities. The primary reason for introducing them is to simplify the statements of several theorems. 2.1 The extended real line The extended real line is obtained by a largest element ∞, and a smallest element −∞, to the real line R: R∗ = R ∪ {−∞, ∞} (2.1) Here ∞ and −∞ are abstract elements. We extend the order relation to R by declaring that −∞ < x < ∞ for all x ∈ R (2.2) Much of our work will be on R∗ , instead of just R. We define addition on R∗ as follows: x + ∞ = ∞ = ∞ + x for all x ∈ R∗ with x > −∞ (2.3) y + (−∞) = −∞ = (−∞) + y for all y ∈ R∗ with y < ∞. (2.4) Note that ∞ + (−∞) is not defined, 23 24 Ambar N. Sengupta i.e. there is no useful or consistent definition for it. The following algebraic facts continue to hold in R∗ : x + y = y + x, x + (y + z) = (x + y) + z, (2.5) whenever either side of these equations holds (i.e. if one side is defined then so is the other and the equality 2.2 Neighborhoods A neighborhood of a point p ∈ R is an interval of the form (p − δ, p + δ) where δ > 0 is any positive real number. Thus, the neighborhood consists of all points distance less than δ from p: (p − δ, p + δ) = {x ∈ R : |x − p| < δ}. (2.6) For example, (1.2, 2.8) is a neighborhood of 2. A typical neighborhood of 0 is of the form (−ε, ε) for any positive real number ε. A neighborhood of ∞ in R∗ = R ∪ {−∞, ∞} is a ray of the form (t, ∞] = {x ∈ R∗ : x > t} with t any real number. For example, (5, ∞] is a neigborhood of ∞. A neighborhood of −∞ in R∗ is a ray of the form [−∞, s) = {x ∈ R∗ : x < s}  Notes in Introductory Real Analysis 25 where s ∈ R. An example is [−∞, 4) Observe that if U and V are neighborhoods of p then U ∩V is also a neighbor- hood of p. In fact, either U contains V as a subset of vice versa, and so U ∩V is just the smaller of the two neighborhoods. Observe also that if N is a neighborhood of a point p, and if q ∈ N then q had a neighborhood lying entirely inside N. For example, the neighborhood (2, 4) of 3 contains 2.5, and we can form the neighborhood (2, 3) of 2.5 lying entirely inside (2, 4). Here is a simple but fundamental observation: Distinct points of R∗ have disjoint neighborhoods. (2.7) This is called the Hausdorff property of R∗. For example, 3 and 5 have the neighborhoods (2, 4) and (4.5, 5.5) The points 2 and ∞ have disjoint neighborhoods, such as (−1, 5) and (12, ∞] Exerise Give examples of disjoint neighborhoods of (i) 2 and −4 (ii) −∞ and 5 (iii) ∞ and −∞ (iv) 1 and −1 2.3 Types of points for a set Consider a set S ⊂ R∗. A point p ∈ R∗ is said to be an interior point of S if it has a neighborhood U lying entirely inside S, i.e. with U ⊂ S.  26 Ambar N. Sengupta For example, for the set E = (−4, 5] ∪ {6, 8} ∪ [9, ∞], the points −2, 3, 11 are interior points. The point ∞ is also an interior point of E. A point p is an exterior point if it has a neighborhood U lying entirely outside S, i.e. U ⊂ Sc. For example, for the set E above, points −5, 7, and −∞ are exterior to E. A point which is neither interior to S nor exterior to S is a boundary point of S. Thus p is a boundary point of S if every neighborhood of p intersects both S and Sc. In the example above, the boundary points of E are −4, 5, 6, 8, 9, ∞. Next consider the set {3} ∪ (5, ∞) The boundary points are 3, 5, and ∞. It is important to observe that if we work with the real line R instead of the extended line R∗ then we must exclude ∞ as a boundary point, because it doesn’t exist as far as R is concerned. Example For the set A = [−∞, 4) ∪ {5, 9} ∪ [6, 7), decide which of the following are true and which false: (i) −6 is an interior point (T) (ii) 6 is an interior point (F) (iii) 9 is a boundary point (T) (iv) 5 is an interior point (F) Exerise For the set B = [−∞, −5) ∪ {2, 5, 8} ∪ [4, 7), decide which of the following are true and which false: (i) −6 is an interior point (ii) −5 is an interior point (iii) 5 is a boundary point (iv) 4 is an interior point (v) 7 is a boundary point.  Notes in Introductory Real Analysis 27 2.4 Interior, Exterior, and Boundary of a Set The set of all interior points of a set S is denoted S0 and is called the interior of S. The set of all boundary points of S is denoted ∂S and is called the boundary of S. The set of all points exterior to S is the exterior of S, and we shall denote it Sext. Thus, the whole extended line R∗ is split up into three disjoint pieces: R∗ = S0 ∪ ∂S ∪ Sext (2.8) Recall that a point p is on the boundary of S if every neighborhood of the point intersects both S and Sc. But this is exactly the condition for p to be on the boundary of Sc. Thus ∂S = ∂Sc. (2.9) The interior of the entire extended line R∗ is all of R∗. So ∂R∗ = 0. / Example For the set A = [−∞, 4) ∪ {5, 9} ∪ [6, 7), (i) A0 = [−∞, 4) ∪ (6, 7) (ii) ∂A = {4, 5, 9, 7, 6} (iii) Ac = [4, 5) ∪ (5, 6) ∪ [7, 9) ∪ (9, ∞] (iv) the interior of the complement Ac is (Ac )0 = (4, 5) ∪ (5, 6) ∪ (7, 9) ∪ (9, ∞]  28 Ambar N. Sengupta For the set G = (3, ∞) the boundary of G, when viewed as a subset of R∗ , is ∂G = {3, ∞}. But if we decide to work only inside R then the boundary of G is just {3}. Exerise For the set B = {−4, 8} ∪ [1, 7) ∪ [9, ∞), (i) B0 = (ii) ∂B = (iii) Bc = (iv) the interior of the complement Bc is (Bc )0 = 2.5 Open Sets and Topology We say that a set is open if it does not contain any of its boundary points. For example, (2, 3) ∪ (5, 9) is open. Also (4, ∞] is open. But (3, 4] is not open, because the point 4 is a boundary point. The entire extended line R∗ is open, because it has no boundary points. Moreove, the empty set 0/ is open, because, again, it doesn’t have any boundary points. Notice then that every point of an open set is an interior point. Thus, a set S is open means that S0 = S.  Notes in Introductory Real Analysis 29 Thus for an open set S each point has a neighborhood contained entirely inside S. In other words, S is made up of a union of neighborhoods. Viewed in this way, it becomes clear that the union of open sets is an open set. Now consider two open sets A and B. We will show that A ∩ B is open. Take any point p ∈ A ∩ B. Then p is in both A and B. Since p ∈ A and since A is open, there is a neighborhood U of p which is a subset of A: U ⊂ A. Similarly there is a neighborhood V of p which is a subset of B: V ⊂B But then U ∩V is a neighborhood of p which is a subset of both A and B: U ∩V ⊂ A ∩ B. Thus every point in A ∩ B has a neighborhood lying inside A ∩ B. Consequently, A ∩ B is open. Now consider three open sets A, B,C. The intersection A ∩ B ∩C can be viewed as (A ∩ B) ∩C But here both A ∩ B and C are open, and hence so is their intersection. Thus, A ∩ B ∩C is open. This type of argument works for any finite number of open sets. Thus: The intersection of a finite number of open sets is open. Exerise Check that the intersection of the sets (4, ∞) and (−3 5) and (2, 6) is open. The collection of all open subsets of R is called the topology of R. The set of all open subsets of R∗ is called the topology of R∗. 30 Ambar N. Sengupta 2.6 Closed Sets A set S is said to be closed if it contains all its boundary points. In other words, S is closed if ∂S ⊂ S Thus, [4, 8] ∪ [9, ∞] is closed. But [4, 5) is not closed because the boundary point 5 is not in this set. The set [3, ∞) is not closed (as a subset of R∗ ) because the boundary point ∞ is not inside the set. But, viewed as a subset of R it is closed. So we need to be careful in deciding what is close and what isn’t: a set may be closed viewed as a subset of R but not as a subset of R∗. The full extended line R∗ is closed. The empty set 0/ is also closed. Note that the sets R∗ and 0/ are both open and closed. 2.7 Open Sets and Closed Sets Consider a set S ⊂ R∗. If S is open then its boundary points are all outside S: ∂S ⊂ Sc. But recall that the boundary of S is the same as the boundary of the complement Sc. Thus, for S to be open we must have ∂(Sc ) ⊂ Sc , which means that Sc contains all its boundary points. But this means that Sc is closed. Thus, if a set is open then its complement is closed. The converse is also true: if a set is closed then its complement is open. Thus, Notes in Introductory Real Analysis 31 Theorem 8 A subset of R∗ is open if and only if its complement is closed. Exercise. Consider the open set (1, 5). Check that its complement if closed. Exercise. Consider the closed set [4, ∞]. Show that its complement is open. 2.8 Closed sets in R and in R∗ The set [3, ∞) is closed in R, but is not closed in R∗. This is because in R it has only the boundary point 3, which it contains; in contrast, in R∗ the point ∞ is also a boundary point and is not in the set. Thus, when working with closed sets it is important to bear in mind the distinction between being closed in R and being closed in R∗. There is no such distinction for open sets. 2.9 Closure of a set The closure of a set S is obtained by throwing in all its boundary points: S = S ∪ ∂S (2.10) Of course, if S is closed then its closure is itself. For example, the closure of (3, 5) is [3, 5]. The closure of (3, ∞) is [3, ∞]. (But, in R the closure of (3, ∞) is [3, ∞).) Let us see what the closure of Q is. Now every point in R∗ is a boundary point of Q: ∂Q = R∗ , because any neighborhood of any point contains both rationals (points in Q) and irrationals (points outside Q). It is useful to think of the closure S = S ∪ ∂S in this way: A point p is in S if and only if every neighborhood of p intersects S. 32 Ambar N. Sengupta 2.10 The closure of a set is closed Consider the closure S of a set S. We will show that S is a closed set. Take any boundary point p ∈ ∂S. We have to show that p is actually in S. Now let N be any neighborhood of p. Then N contains a point q in S. Choose (as we may) a neighborhood W of q lying entirely inside N. Since q ∈ S it follows that W contains a point of S. Thus, N contains a point of S. So we have seen that every neighborhood of p contains a point of S. This means p ∈ S. Thus, we have shown that every boundary point of S is in S, and so S is closed. 2.11 S is the smallest closed set containing S Consider any closed set K with S⊂K Take any p ∈ S. Then every neighborhood of p intersects S, and hence also K. Thus every neighborhood of p intersects K. So, either p is in the interior of K or on its boundary. But K is closed, and so in either case p is in K. Thus, S⊂K What we hae shown then is that S is the smallest closed set containing S as a subset. 2.12 R∗ is compact There is a special property of R∗ whose full significance is best appreciated at a later stage. However, we have the language and tools to state and prove it now and shall do so. An open cover of R∗ is a collection U of open sets whose union covers all of R∗. For example, {[−∞, 5), (−1, 8), (7, 9) ∪ (11, 15), (8, ∞]} is an open cover of R∗. It is best to draw a picture for yourself to make this clear. More formaly, an open cover of R∗ is a set U of open subsets of R∗ such that every point x ∈ R∗ falls inside some open set U in the collection U , i.e. for each x ∈ R∗ there exists U ∈ U such that x ∈ U ∈ U. Notes in Introductory Real Analysis 33 Another example of an open cover of R∗ is {[−∞, 50), (2, 100), (8, 200) ∪ (701, 800), (150, 750), (760, ∞]}. The examples of open covers of R∗ given above are both finite collections of sets. Let us look at an example of a cover which uses infinitely many sets. We start with an open cover of R, as opposed to R∗ : take all intervals of the form (a, a + 2), where a runs over all integers:..., (−8, −6), (−7, −5),...(−2, 0), (−1, 1), (0, 2), (1, 3),... In short we are looking at the collection U 0 = {(a, a + 2) : a ∈ Z}, where, recall that Z is the set of all integers. This collection fails to include −∞ and ∞. To obtain an open cover of R∗ from U 0 we could, for example, just throw in the open sets [−∞, −4) and (5, ∞]. Thus, this gives us an open cover of R∗ : U = {[−∞, −4), (5∞]} ∪ {(a, a + 2) : a ∈ Z}. If you look at this, however, you realize that even though this collection does contain infinitely many open sets, we don’t reall need all of them to cover R∗. Indeed, we could just use the sub-collection V = {[−∞, −4), (−3, −1), (−2, 0), (−1, 1), (0, 2), (1, 3), (2, 4), (3, 5), (4, 6), (5, ∞]} and this would cover all of R∗. This is not just a feature of one particular example. It turns out that Every open cover of R∗ has a finite subcover. This property is called compactness, and so Theorem 9 R∗ is compact. We can prove the compactness of R∗ using the completeness of the real line. Proof of compactness of R∗. Let U be an open cover of R∗. This is a collection of open sets such that every point of R∗ falls inside some set in the collection. In particular, −∞ is in some open set W ∈ U. Thus W contains a neighborhood of −∞, i.e. [−∞,t) ⊂ W 34 Ambar N. Sengupta for some t ∈ R. So, for one thing, all the elements of R∗ less than t are covered by just the one set W in U ; we would not need any other set from the cover if we are to stick to points to the left of t. Now let S = {x ∈ R∗ : finitely many sets in U cover [−∞, x]} This set is not empty because −∞ ∈ S. In fact, t − 1 is also in S. By completeness, the set S has a supremum sup S. Let m = sup S. We will argue in three steps: (i) First we explain that m isn’t −∞. (ii) Next we prove that m is in fact, ∞. (iii) Finally we prove that [∞, m], i.e. all of R∗ , can be covered by just finitely many sets in U. For (i), note, as before, that there is a real number t such that t − 1 is in S, and so m (being an upper bound of S) has to be ≥ t − 1. In particular, m is certainly not −∞. Now we show that m = ∞. Suppose to the contrary that m < ∞. Now there is some open set U ∈ U such that m ∈ U, because U covers all points of R∗. Since m is not −∞, and is being assumed to be also not ∞, it is in R and so there is some neighborhood (m − ε, m + ε) ⊂ U, where ε is a positive real number. Now m − ε being less than the least upper bound m of S, there has to be some x ∈ S which is greater than m − ε. Thus m − ε < x ≤ m for some x ∈ S. So, x being in S, there are finitely many sets, say U1 ,...,UN , in U , which cover the ray segment [−∞, x]. The set U covers (m − ε, m + ε). Thus, the collection {U1 ,...,UN ,U} covers all of [−∞, m + ε). Notes in Introductory Real Analysis 35 But this means that, for example, m + 21 ε is in S, for the ray segment [−∞, m + 1/2] is covered by the finite collection U1 ,...,UN ,U. But now we have a contradiction, because we have found a number, m + ε/2, greater than m, which is in S. Thus, our original hypothesis concerning m must have been wrong. So m = ∞. Finally we prove that R∗ is covered by finitely many sets in U. The element ∞ ∈ R∗ falls inside some open set V in the collection U. Therefore, there is some ‘ray-neighborhood’ of ∞ (r, ∞] ⊂ V, for some real number r. Now r being less than m = ∞, and the latter being the least upper bound of S, there must be some y ∈ (r, ∞] which is in S. Thus, [−∞, y] is covered by finitely many open sets, say V1 ,...,Vk , in U. Note that [−∞, y] ∪ (r, ∞] = R∗. But then V1 ,...,Vk ,V cover all of [−∞, ∞], since V covers the segment (r, ∞]. This prove that finitely many sets from U cover all of R∗. 2.13 Compactness of closed subsets of R∗. Consider now any closed subset D of R∗. We will prove that it is compact, i.e. that any open cover of D has a finite sub-cover. Consider any open cover U of D. This is a collection of open sets such that every point of D is covered by some set in the collection. Now the set Dc , being the complement of a closed set, is an open set. Throw this into the collection, and consider U 0 = U ∪ {Dc }. This covers all of R∗ : any point of D would be covered by a set in U while any point in Dc is, of course, covered by Dc. Then we know that there has to be a finite sub-collection V 0 ⊂ U0 36 Ambar N. Sengupta which covers R∗. Now throw out Dc from V 0 in case it is there, and consider the collection V = V 0 \ {Dc }. Of course, this is a finite subcollection of U. Moreover, it covers all points of D, because no point in D could have been covered by the ‘rejected’ set Dc. Thus, D is covered by a finite sub-collection of sets from U. Thus, every closed subset of R∗ is compact. The converse is also true, and we can state: Theorem 10 A subset of R∗ is compact if and only if it is closed. We will leave out the proof of the converse part of this result. 2.14 The Heine-Borel Theorem: Compact subsets of R A subset B of R is said to be bounded if B ⊂ [−N, N] for some real number N. Thus, for B to be bounded, there should exist a real number N such that |x| < N for all x ∈ B. Recall that a subset of R is closed in R if it contains no boundary point. Equiv- alently, if a subset of R is closed if its complement in R is open. Consider, for instance, the set [4, ∞) ⊂ R. This is a closed subset of R because, in R, its only boundary point is 4, and this point lies in the set. Note that [4, ∞) is not closed when considered as a subset of R∗. We can now state the Heine-Borel theorem: Theorem 11 Every closed and bounded subset of R is compact. Conversely, ev- ery compact subset of R is closed and bounded. Notes in Introductory Real Analysis 37 We shall prove half of this. Suppose K ⊂ R is closed and bounded as a subset of R. Boundedness implies that K ⊂ [−N, N] for some real number N. Then K is closed also as a subset of R∗. Let’s check this. We will show that the complement U of K in R∗ is open. Consider any point p ∈ U. If p ∈ R then we already know, by closedness of K, that p has a neighborhood lying inside U. If p = ∞ then the neighborhood of p given by (N + 1, ∞] lies entirely outside K. If p = −∞ then the neighborhood [−∞, −N − 1) is entirely outside K. Thus, in all cases, p has a neighborhood outside K in R∗. So the complement of K in R∗ is open, and hence K is closed in R∗. We know that then K must be compact. 2.15 Sequences A sequence of elements in a set A is a string a1 , a2 , a3 ,... of elements of A. More precisely, the sequence is actually a mapping a : P → A : n 7→ an. We will often be concerned with sequences in R∗. Sometimes out sequence will be specified explicitly as a string of numbers; for example, −5, −3, 5, 6, 9, 12,... Sometimes we may have a formula for the n-th term: (−1)n an =. n+1 38 Ambar N. Sengupta Sometimes we have a sequence specified recursively. For example, we might know that b1 = 1, b2 = 1 and then bn = bn−1 + bn−2 , for all n ≥ 3. This generates the Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21,... Sometimes it is better to label a sequence starting with the index 0: a0 , a1 , a2 ,... For example, for a fixed real number r we can form the sequence r0 = 1, r1 = r, r2 , r3 ,... 2.16 Limits points of a sequence Consider a sequence x1 , x2 , x3 ,.... in R∗. A point p ∈ R∗ is said to be a limit point of this sequence if every neigh- borhood of p is visited infinitely often by the sequence. For example, for the sequence 1 1 1 1 1 1 1 1 1 , 4, , 4+ , , 4+ , , 4+ , , 4 + ,... 2 3 3 4 4 5 5 6 6 it is intuitively clear that both 0 and 4 are limit points. Suppose a sequence (xn ) lies entirely inside a set S ⊂ R: xn ∈ S, for all n ∈ P Consider then any limit point p of this sequence. Let U be any neighborhood of p. We know that this neighborhood is visited infinitely often by the sequence. Therefore, at least one point of S must be in U. Thus, every neigborhood of p contains a point of S, and so p ∈ S. Thus, any limit point of a sequence which lies always in a set S must be in the closure of S. Notes in Introductory Real Analysis 39 2.17 Bolzano-Weierstrass Theorem Consider any sequence in R∗. We shall prove that it must have at least one limit point. Suppose to the contrary that no point is a limit of the given sequence. Then each point p of R∗ has a neighborhood U p which is visited only finitely often by the sequence. The neighborhoods U p form an open cover of all of R∗. Then, by compactness of R∗ , there are finitely many of them, say U p1 ,...,U pN which cover all of R∗. But this is impossible: we have covered all of the extended real line R∗ with finitely many sets each of which is visited only finitely many times by the sequence! Thus, we have a contradiction and so there exist a limit of the sequence. Notice that all we needed above was the compactness of R∗. Thus, we have the more general Bolzano-Weierstrass theorem: Theorem 12 Every sequence in a compact set has at least one limit point. Of course, we have seen that a sequence may well have more than one limit point. For example, the sequence, 1, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5,... visits every natural number infinitely often and so every natural number is a limit point of the sequence. In the next subsection we will show how to actually obtain a limit point of a sequence. 2.18 Limit points and suprema and infima Consider a sequence x1 , x2 , x3 ,... Let p be any limit point of this sequence. Now let S1 be the least upper bound of the set {x1 , x2 ,...}: S1 = sup{x1 , x2 ,...} 40 Ambar N. Sengupta In particular, S1 ≥ xn for each n. Then it is clear that no point ‘to the right’ of S1 could possibly be a limit of the sequence: indeed any point r > S1 would have a neighborhood lying entirely to the ‘right’ of S1 and this would never be visited by the sequence. Thus, p cannot be > S1. So p ≤ S1 Now we can apply the same argument to S2 = sup{x2 , x3 , x4 ,...} Again, p could not be greater than S2 for then it would have a neighborhood to the right of S2 and this would have to be visited infinitely often by x1 , x2 ,... and so at least once by x2 , x3 ,.... Thus, p ≤ S2 In this way, we can form S3 = sup{x3 , x4 ,...} and have again p ≤ S3. Thus, p is a lower bound for all the Sk ’s: p ≤ Sk for all k. Now the inf of a set is the greatest lower bound of the set. Therefore, p ≤ inf{S1 , S2 , S3 ,...}. The infemum on the right is an important object and is called the limsup of the given sequence: def lim supn→∞ xn = inf{supn≥k xn } : k ∈ P} (2.11) So we have proved that Every limit point is ≤ the limsup Notes in Introductory Real Analysis 41 for any sequence. Similarly, we have the notion of liminf: it is the supermum of the sequence of infemums: def lim infn→∞ xn = sup{infn≥k xn } : k ∈ P} (2.12) In just the same way as before, we can prove Every limit point is ≥ the limsup Thus, for any limit point p if any sequence (xn ) we have lim infn→∞ xn ≤ any limit point ≤ lim supn→∞ xn. (2.13) In fact, the limsup and liminf of the sequence are also limit points, but we won’t prove this here. 2.19 Limit of a sequence A sequence (sn ) is said to lie in a set S eventually if after a certain value of n, all the sn lie in S. Put another way, we say that sn lies in S for large n, if sn ∈ S for all values of n beyond some value, say n0. For example, the sequence −5, −4, −3, −2, −1, 0, 1, 2, 3,... will lie in the set (20, ∞) eventually. Let us note again: a sequence (sn ) lies in a set S eventually if there is an n0 ∈ P such that sn ∈ S for all n ∈ P with n ≥ n0. A sequence (xn ) in R∗ is said to have limit L ∈ R∗ if for any neighborhood U of L the sequence lies in this neighborhood eventually. We denote this symbolically as xn → L, as n → ∞. 42 Ambar N. Sengupta We shall see later that if a limit exists then it is unique; the limit L is denoted lim xn. n→∞ The notion of limit is one of the central notions in mathematics. For example, the sequence 1, 2, 3, 4,.... has limit ∞. For, if we take any neighborhood of ∞, say (t, ∞] then eventually n will exceed t (by the archimidean property) and so the sequence will stay in (t, ∞] from the n-th term onwards. The sequence 1 1 1 1, , , ,... 2 3 4 has limit 0. We will look at this more carefully soon. 2.20 Simple Examples of limits Let us look at some examples of sequences and try to see what their limits are. The simplest sequence is a constant sequence which keeps repeating the same value. For example, 3, 3, 3, 3,.... The limit of the this sequence is 3: clearly every neighborhood of 3 is hit even- tually by the sequence (indeed it is hit every time, since the sequence is stuck at 3). The sequence −1, 4, 5, 7, 8, 8, 8, 8, 8,.... which eventually stabilizes at the constant value 8 has limit 8. For, again, given any neighborhood of 8 the sequence falls inside this neighborhood eventually and stays there. In contrast, the sequence 1, 3, 4, 1, 3, 4, 1, 3, 4, 1, 3, 4,... Notes in Introductory Real Analysis 43 does not have a limit. For example, the point 3 cannot be the limit of the sequence because, for instance, (2.5, 3.5) is a neighborhood of 3, but the sequence keeps falling outside this neighborhood (when it hits 1 or 4). The sequence 1, 3, 5, 7,... has limit ∞. If you take any neighborhood of ∞, an interval of the form (t, ∞] then eventually the sequence falls inside the nighborhood and stays in there. 2.21 The sequence 1/n Consider again the sequence 1, 1/2, 1/3, 1/4,... It is intuitively clear that this sequence has limit 0. But let us prove this. Note first that the n-th term of the sequence is xn = 1/n. We have to show that given any neighborhood of 0, our sequence will eventu- ally lie inside this neighborhood. So consider any neighborhood of 0: (−ε, ε), where ε is a positive real number. We have to show that xn lies in (−ε, ε) for all n beyond some value. Thus we should show that 1 1. 44 Ambar N. Sengupta The archimedean property guarantees the existence of an n0 ∈ P for which n0 ε > 1. Therefore, nε > 1 for all integers n ≥ n0. This proves that the sequence does indeed tend to the limit 0: 1 → 0, as n → ∞ n 2.22 The sequence Rn Consider the sequence 30 , 31 , 32 , 33 ,... It is clear that in this sequence the terms get large very quickly, and intuitively it is clear that the sequence has limit ∞. Consider now the sequence of powers Rn where R > 1. Let us see by how much each term exceeds the preceding: Rn − Rn−1 = Rn−1 (R − 1) But R is > 1, and so the multiplier Rn−1 is ≥ 1 (it is equal to 1 when n is actually 1). Thus, Rn − Rn−1 ≥ R − 1. Let us write x for R − 1, and note that x>0 and we have Rn − Rn−1 > x. Now it takes n ‘steps’ to climb from R0 to Rn , and so Rn ≥ 1 + nx, Notes in Introductory Real Analysis 45 because each step is at least x. Now it is clear that Rn → ∞ as n → ∞. We just have to show that for any given t, 1 + nx > t when n is large enough. But this is just Archimedes again: some multiple n0 x of xexceeds t − 1, and then nx > t − 1 for all n ≥ n0. Note again that it is important that x > 0, and this comes from the fact that R > 1. Thus, lim Rn = ∞ for all R > 1. n→∞ Now consider the case R = 1. In this case the sequence is just all powers of 1, and so it is 1, 1, 1, 1,.... Thus lim Rn = 1 if R = 1. n→∞ Next consider the case 0 < R < 1. As an example, we have R = 1/3 and the sequence 1 1 1 1, , 2 , 3 ,... 3 3 3 We have here a sequence with denomiantor going to ∞ and numerator fixed at 1. It seems then clear that the sequence goes to 0. Indeed, for any real ε > 0 we just have to wait till 3n > 1/ε and this would ensure that 1 < ε, 3n and so 1 ∈ (−ε, ε) for large n. 3n Now consider the general case of Rn 46 Ambar N. Sengupta where R > 1. Observe that 1 R= , r where r = R−1 is > 1. Thus 1 Rn = , rn where r > 1. Now, as we have seen above, the denominator rn goes to infinity, and so it seems clear that 1/rn should decrease to 0. So we will prove that Rn → 0 in this case. Consider any neighborhood of 0: (−ε, ε) where ε > 0 is a positive real number. We want Rn to be in this neighborhood, i.e. Rn should be < ε. But this means we should show that rn is > 1/ε for all n large enough. But we know that rn → ∞ So, after some n0 , we have rn > ε−1 for all n ≥ n0. Consequently, 1 Rn = t: xn0 > t. But recall that we are dealing with a monotone increasing sequence. Conse- quently, xn > t for all n ≥ n0. But recall that L is an upper bound of {x1 , x2 ,...}; therefore, all the xn are ≤ t. Those xn which are > t but ≤ L are, of course, in the neighborhood U of L. Thus, xn ∈ U for all n ≥ n0. This proves that L is indeed the limit of the sequence (xn ). In a similar way, if xn is a monotone decreasing sequence then lim xn = inf xn. n→∞ n≥1 2.24 The limit of a sequence is unique Now we shall prove that a sequence can have at most one limit. Consider a sequence (xn ) and suppose both L and L0 are limits of the sequence, and L 6= L0. We will arrive at a contradiction. Since L and L0 are distinct, they have disjoint neighborhoods U and U 0 respectively. Since xn → L we know that xn ∈ U eventually. But xn → L0 , and so xn ∈ U 0 eventually. But this is impossible since U and U 0 are disjoint and thus have no element in common. Thus, a sequence which has a limit must have a unique limit. 2.25 Convergent sequences and Cauchy sequences A sequence (xn ) in R is said to be convergent if it has a limit and the limit is a real number. If xn → L, and L ∈ R, we also say that the sequence xn converges to L. Note that xn → L ∈ R if for any real r > 0 there is an n0 ∈ P such that |xn − L| < r 50 Ambar N. Sengupta for all natural numbers n > n0. This has a consequence: since the xn ’s are all accumulating to L they are also getting close to each other. More precisely, for any ε > 0 we can find N ∈ P such that |xn − xm | < ε for all natural numbers n, m > N. To prove this, observe that there is some N ∈ P such that |xk − L| < ε/2 for all k ∈ P with k > N. But then for any n, m ∈ P with n, m > N we have |xn − xm | = |xn − L + L − xm | ≤ |xn − L| + |L − xm | = |xn − L| + |xm − L| < ε/2 + ε/2 = ε. A sequence (xn ) in R which bunches up on itself in the sense above is said to be a Cauchy sequence, i.e. if for any δ > 0 there is an N ∈ P such that |xn − xm | < ε for all natural numbers n, m > N. 2.26 Every Cauchy sequence is bounded Consider a Cauchy sequence (xn ) in R. Then, eventually the points of this se- quence are at most distance 1 from each other; in fact, there is an N ∈ P such that |xn − xm | < 1 for all natural numbers n, m > N. In particular, fixing a particular j > N we have xn ∈ (x j − 1, x j + 1), for all n ≥ N. So x j − 1 < xn < x j + 1 for all n ∈ {N + 1, N + 2,...}. Thus, at least from the (N + 1)-th term on, the sequence is bounded. But the terms not counted here, x1 ,..., xN Notes in Introductory Real Analysis 51 are just finitely many, and so they have a largest B and a smallest A among them. Thus, every xn is ≤ max{B, x j + 1} and every xn is ≥ min{B, x j − 1} Thus the entire sequence is bounded. 2.27 Every Cauchy sequence is convergent The completeness property of the real line R has an equivalent formulation: Theorem 13 Every Cauchy sequence in R converges. Let us prove this. Consider a Cauchy sequence (xn ) in R. We have seen that it is bounded. Thus, a ≤ xn ≤ b for all n ∈ P for some real numbers a and b. We know that a sequence always has a limit point in R∗. Let L be a limit point of (xn ). We will prove that L is a real number and xn → L. First it is clear that L must also be in [a, b]. Therefore, L is not ∞ or −∞, and is a real number. Take any neighborhood (L − δ, L + δ) of L, where δ > 0 is a real number. The sequence (xn ) visits the neighborhood δ δ (L − , L + ) 2 2 infinitely often. Now we also know that eventually, the terms of the sequence vary from each other by < δ/2, i.e. there is some N ∈ P such that |xn − xm | < δ/2 for all n, m ∈ P with n, m > N. 52 Ambar N. Sengupta We can choose some j > N such that δ δ x j ∈ (L − , L + ) 2 2 because the sequence visits this neighborhood infinitely often. Then |xn − L| = |xn − x j + x j − L| ≤ |xn − x j | + |x j − L| δ δ < + 2 2 = δ. This means that xn ∈ (L − δ, L + δ) for all n > N. Thus, xn → L. 2.28 The rationals are countable A set is set to be countable if it is finite or if its elements can be enumerated in a sequence. Thus, S is countable if there is a sequence x1 , x2 ,... such that S = {x1 , x2 , x3 ,...} For example, the even numbers form a countable set: {2, 4, 6, 8,...} It may seem at first that the set Z of integers is not countable, but we can certainly lay out all the integers in a sequence: 0, 1, −1, 2, −2, 3, −3,... It is much harder to see that the rationals Q are also countable. This is what we shall prove now. We will constuct a sequence which enumerates all the rationals, i.e. a sequence r1 , r2 ,... such that {r1 , r2 , r3 ,...} = Q. Notes in Introductory Real Analysis 53 Put another way, we will encode each rational by a unique natural number; thus to each natural number n we would associate a rational rn , and in this way we will cover all rationals. There can be many such encoding schemes. Here is one: Take any rational x ≥ 0 and write it as qp where p and q are non-negative integers (of course q ≥ 1) and q is the smallest possible denominator among all such ways of writing x. (For example, 0.6 can be written as both 6/10 and 3/5, but we would take 3/5 as our representation.) We know that every non-negative rational can be represented in this way uniquely. Associate to this x the natural number f (x) = 2q × 3 p For example, f (0) = f (0/1) = 21 × 30 = 2 f (1) = f (1/1) = 21 31 = 6, f (4/5) = 25 × 34 = 32 × 81 = 2592. This associates a unique natural number f (x) to each non-negative rational x. If x ∈ Q is negative, x < 0, let us define f (x) to be just 5 times f (−x): f (x) = f (−x) × 5 if x < 0 Now we have labeled each rational by a unique natural number. To enumerate the rationals in a sequence all we have to do then is to run this in reverse: let r1 be the rational number x with the smallest value for f (x) (so r1 is in fact 0 because f (0) = 2 is the lowest value possible for f ); let r2 be the rational number with the next lowest value for f (x), and so on. Thus, rn is the rational for which f (rn ) is the first value of f (x) greater than f (r1 ),..., f (rn−1 ). For example, r1 = 0 and r2 = 1 because f (1) = 21 31 = 6, the next value for f (x) after 2 Thus we have produced a sequence of distinct elements r1 , r2 ,... such that {r1 , r2 ,...} = Q 54 Ambar N. Sengupta 2.29 The real numbers are uncountable We will prove that R is not countable, i.e. there is no sequence which touches on all the real numbers. In fact we will show that even the real numbers between 0 and 1 cannot be enumerated in a sequence. The argument used here is the celebrated diagonal method due to Georg Can- tor (1845-1912). The strategy is to make a list of strings, then take the main diagonal string, and then form a new string by altering each element of the diago- nal. Consider any sequence x1 , x2 ,... lying in (0, 1). Now each real number y in (0, 1) can be expressed uniquely in the form y1 y2 y3 y = 0.y1 y2 y3... = + 2 + 3 +··· , 10 10 10 where y1 , y2 ,... ∈ {0, 1, 2, 3,..., 9} and we exclude all such representations which use an infinite string of 9’s at the end (for example, instead of 0.19999.... we would use 0.20000....). Thus each of the numbers x1 , x2 , also has such an expansion: x1 = 0.x11 x12 x13... x2 = 0.x21 x22 x23... x3 = 0.x31 x32 x33...... = ··· Now form a number w = 0.w1 w2 w3... as follows: take w1 to be any number in {1, 2,..., 8} other than x11 ; then choose w2 ∈ {1, 2,..., 8} other than x22 , and so on. This way we make the n-th decimal place of w different from the n-th decimal place of xn , for every n. Then w cannot be equal to any of the xn , and w ∈ (0, 1). Thus the original sequence cannot possibly have covered all of (0, 1). [The reason we excluded 0 and 9 from our choices for wn was to avoid ending up at 0 or 1.] Notes in Introductory Real Analysis 55 2.30 Connected Sets Consider a subset S of R∗. If U is an open set then the part of U in S, i.e. U ∩ S is said to be open in S. For example, in [2, 5] the set [2, 4) is open because, for example, [2, 4) = (1, 4) ∩ [2, 5] Put another way, a subset J of S is said to be open in S if every p ∈ J has a neighborhood N such that every point of N which is in S is in fact in J, i.e. N ∩J ⊂ S A set S is said to be connected if it cannot be split into two non-empty disjoint pieces A and B each of which is open in S. The main result for connected sets is: Theorem 14 A subset of R, or of R∗ , is connected if and only if it is an interval. 56 Ambar N. Sengupta Chapter 3 Integration Modern integration theory was built from the works of Newton, Riemann, and Lebesgue, but the origins of integration theory lie in the computation of areas and volumes. The idea of measuring area of a curved region by lower and upper bounds is present even Archimedes’ study of the area enclosed by a circle. The area of a rectangle whose sides are 2 units and 3 units is 6 square units, because if you tile the rectangle with unit squares you will need three rows of such squares, each row having 2 tiles. By extension, the area of a rectangle of sides 1/4 unit and 1/5 unit should be 1/20 square units because we would need 20 of these rectangles to cover a unit square. Thus, it is clear that the area enclosed by a rectangle whose sides are a units and b units, where a and b are rational numbers, is ab square units. Consider now a rectangle R whose sides a and b are possibly not rational. Take any rationals a0 , a00 , and b0 , b00 with a0 < a < a00 , and b0 < b < b00. Then a rectangle R0 of sides a0 by b0 sits inside R, while a rectangle R00 of sides a00 by b00 contains R. Thus, it makes sense to suppose that area of R0 ≤ area of R ≤ area of R00 which is to say: a0 b0 ≤ area of R ≤ a00 b00 It is intuitively clear, and not hard to prove, that ab is the unique real number that lies between all the possible values of a0 b0 and a00 b00. Thus, area of R = ab 57 58 Ambar N. Sengupta The rectangle R is made up of two congruent right angled triangles, and so each of these would have area (ab)/2. This makes it possible to compute the areas of all kinds of polygonal figures but cutting up these fgures into right angled trian- gles. However, this strategy fails when we try to compute the area enclosed by a circle. No amount of cutting would turn a disc into a finite number of right angled triangles. Archimedes computed area enclosed by a circle C by consider polygons P0 and P00 , where P lies inside the circle C and P00 encloses the circle C; thus area A0 enclosed by P0 ≤ area enclosed by C ≤ area A00 enclosed by P00 Some computation shows that for any ε > 0 there are such polygons P0 and P00 such that A00 − A0 < ε This implies that there is a unique real number A which lies between the area of all the polygons P0 and the polygons P00. Clearly then this number should be the area enclosed by the circle C. Our development of the theory of the Riemann integrals is based on these ideas. 3.1 Approaching the Riemann Integral Consider a function f : [a, b] → R and think of its graph. Assume, for convenience of visualization, that f ≥ 0. Then f specifies a region which lies below its graph and above the x-axis. In general, this is a region whose upper boundary, given by the graph of f , is curved. The integral Z b f a which is also, conveniently, written as Z b f (x) dx a measures the area of this region. The strategy used for computing the area is to cut up the region into vertical slices. The area of each such slice is between the area of a larger ‘upper’ rectangle Notes in Introductory Real Analysis 59 and a smaller ‘lower’ rectangle. Thus we should expect the actual area to be the unique real number lying between these ‘upper rectangle’ areas and the ‘lower rectangle’ areas. 3.2 Riemann Sums We will work with functions on an interval [a, b] ⊂ R ] where < b. A partition X of [a, b] is specified by a sequence X = (x0 , x1 ,..., xN ) of points x0 , x1 ,..., xN ∈ [a, b] with a = x0 < x1 < · · · < xN = b Here N ∈ {1, 2, 3,...}. We will often use the notation def ∆x j = x j − x j−1 (3.1) to denote the length of the j-th interval [x j−1 , x j ] marked out by the partition X. The width or norm of the partition X is the maximum size of these intervals: ||X|| = max ∆x j (3.2) j∈{1,...,N} Consider a function f : [a, b] → R on an interval [a, b] ⊂ R, where a < b. Let M j( f ) = sup f (x) (3.3) x∈[x j−1 ,x j ] 60 Ambar N. Sengupta and m j( f ) = inf f (x) (3.4) x∈[x j−1 ,x j ] Thus, if A j were the area of the region under the graph of f over [x j−1 , x j ] then m j ( f )∆x j ≤ A j ≤ M j ( f )∆x j. Our objective is to specify the actual area A under the graph of f , and this would be the sum of the A j. The upper Riemann sum U( f , X) is defined to be N def U( f , X) = ∑ M j ( f )∆x j (3.5) j=1 and the lower Riemann sum L( f , X) is N def L( f , X) = ∑ m j ( f )∆x j (3.6) j=1 Note that L( f , X) ≤ U( f , X) (3.7) We will show later that in fact every lower sum if less or equal to every upper sum, i.e. L( f , X) is less or equal to U( f ,Y ) for every partitions X and Y of [a, b]. Now consider a sequence X ∗ = (x1∗ ,..., xN∗ ) obtained by picking a point from each interval [x j−1 , x j ]:: x∗j ∈ [x j−1 , x j ] We shall indicate this by writing X∗ < X The Riemann sum S( f , X, X ∗ ) is N ∗ S( f , X, X ) = ∑ f (x∗j )∆x j (3.8) j=1 Notes in Introductory Real Analysis 61 Note that the upper sum is ∞ if and only if one of the M j is ∞, and this occurs if and only if the superemum of f is ∞: U( f , X) = ∞ if and only if sup f (x) = ∞ (3.9) x∈[a,b] However, U( f , X) cannot be −∞, because that would mean that at least one of the M j is −∞ which can only be if f is equal to −∞ on that interval [x j−1 , x j ] which contradicts the fact that f is real-valued. (Note that we are working with non-empty intervals because x j−1 < x j.) Similarly, L( f , X) = −∞ if and only if inf f (x) = −∞ (3.10) x∈[a,b] and L( f , X) can never be ∞. It is useful then to observe that the difference U( f , X) − L( f , X) is always defined, i.e. we never have the ∞ − ∞ situation. 3.3 Definition of the Riemann Integral Consider a function f : [a, b] → R on an interval [a, b] ⊂ R, where a < b. This function is said to be Riemann inte- grable if there is a unique real number I lying between all the lower sums and all the upper sums: L( f , X) ≤ I ≤ U( f , X) (3.11) for every partition X of [a, b]. This number I is the integral of f over [a, b] and denoted Z b Z b f or f (x) dx (3.12) a a The set of all Riemann integrable functions on [a, b] is denote R [a, b] (3.13) 62 Ambar N. Sengupta 3.4 Refining Partitions We work with a function f : [a, b] → R. We use the notation M( f , [s,t]) = sup f (x), and m( f , [s,t]) = inf f (x) (3.14) x∈[s,t] x∈[s,t] Let X = (x0 , x1 ,..., xN ) be a partition of the interval [a, b] and let X 0 be a partition obtained by adding one more point x0 to X. Let us say that x0 lies in the j-th interval x0 ∈ [x j−1 , x j ] Thus, X 0 cuts up [x j−1 , x − j] into two intervals [x j−1 , x0 ] and [x0 , x j ] Let us compare the upper sums U( f , X) and U( f , X 0 ). To this end, let us write M j = M( f , [x j−1 , x j ]), M 0j = M( f , [x j−1 , x0 ]), M 00j = M( f , [x0 , x j ]) It is important to observe that M j ≥ M 0j , and M j ≥ M 00j (3.15) These sums differ only in the contribution that comes from [x j−1 , x j ]: U( f , X) −U( f , X 0 ) = M j (x j − x j−1 ) − M 0j (x0 − x j−1 ) + M 00j (x j − x0 )   = (M j − M 0j )(x0 − x j−1 ) + (M j − M 00j )(x j − x0 ) ≥ 0. Thus U( f , X 0 ) ≤ U( f , X), (3.16) i.e. adding a point to a partition lowers upper sums. If we keep adding points to the initial partition, we keep lower the upper sums. Arguing similarly, it follows that adding points to a partition raises lower sums. This observation is important enough to state as a theorem: Notes in Introductory Real Analysis 63 Theorem 15 Adding points to a partition lowers upper sums and raises lower sums. Thus if X and X 0 are partitions of [a, b], with X 0 containing all the points of X and some more, then L( f , X) ≤ L( f , X 0 ) ≤ U( f , X 0 ) ≤ U( f , X) (3.17) for every function f : [a, b] → R. Using this we can prove that upper sums always dominate lower sums: Theorem 16 If f : [a, b] → R is a function on an interval [a, b] ⊂ R, where a < b, and if X and Y are any partitions of [a, b] then L( f , X) ≤ U( f ,Y ) (3.18) Thus, every lower sum is dominated above by every upper sum. Proof. Here is a useful trick: we can combine the partitions X and Y to form a partition Z which contains all points of X and all the points of Y as well. Therefore, U( f , Z) ≤ U( f ,Y ) and L( f , X) ≤ L( f , Z) Stringing these together we have L( f , X) ≤ L( f , Z) ≤ U( f , Z) ≤ U( f ,Y ) and this gives the desired result. QED We have used the fact that adding a point to a partition lowers upper sums and raises lower sums. It will be useful to take a closer look at this and estimate by how much the upper and lower sums move as points are added to the partition. Recall the formula U( f , X) −U( f , X 0 ) = (M j − M 0j )(x0 − x j−1 ) + (M j − M 00j )(x j − x0 ) (3.19) where X 0 is the partition obtained from X by adding the single point x0 to the j-th interval [x j−1 , x j ] of the partition X = (x0 ,..., xT ). Now suppose instead of adding just the one point x0 , we add m distinct points y1 ,..., ym to the interval [x j−1 , x j ] to form the new partition X 0. We label the new points yi in increasing order y1 < · · · < ym. 64 Ambar N. Sengupta For convenience of notation we write y0 for x j−1 and ym+1 for x j. Thus x j−1 = y0 < y1 < · · · < ym < ym+1 = x j The intervals [yk−1 , yk ] make up the interval [x j−1 , x j ] Then: m+1  0  U( f , X) −U( f , X ) = ∑ M( f , [x j−1 , x j ]) − M( f , [yk−1 , yk ]) (yk − yk−1 ) (3.20) k=1 Now 0 ≤ M( f , [x j−1 , x j ]) − M( f , [yk−1 , yk ]) ≤ 2|| f ||sup , where || f ||sup is supremum of | f | over the full original interval [a, b]. The interval lengths yk − yk−1 add up to ∆x j. Consequently, U( f , X) −U( f , X 0 ) ≤ 2|| f ||sup ∆x j ≤ 2|| f ||sup ||X|| (3.21) This provides an upper bound for how much the upper sum is decreased by addi- tion of points all in one single interval of the original partition X. If we add points to each of N 0 intervals to create a new partition X 0 then U( f , X) −U( f , X 0 ) ≤ 2N 0 || f ||sup ||X|| (3.22) Similarly, L( f , X 0 ) − L( f , X) ≤ 2N 0 || f ||sup ||X|| (3.23) Note that N 0 ≤ N, where N is the total number of new points added. Consequently, we have Lemma 1 If X = (x0 ,..., xT ) is a partition of the interval [a, b] ⊂ R, where a < b, and X 0 is a partition of the same interval containing all the points of X and possibly some more, then U( f , X 0 ) − L( f , X 0 ) ≤ U( f , X) − L( f , X) (3.24) but the decrease in the value of U − L is at most 2N|| f ||sup ||X||: [U( f , X) − L( f , X)] − U( f , X 0 ) − L( f , X 0 ) ≤ 2N|| f ||sup ||X||   (3.25) where N is the number of new points added. Notes in Introductory Real Analysis 65 3.5 The Darboux Criterion Now that we know that the upper sums dominate the lower sums, it is clear that there will be a unique real number between them if and only if the sup of the lower sums equals the inf of the upper sums: Theorem 17 A function f : [a, b] → R is Riemann integrable if and only if sup{U( f , X) : all partitions X of [a, b]} = inf{L( f , X) : all partitions X of [a, b]} (3.26) The common value is ab f. R Thus, the function f is Riemann integrable if and only if there is no ‘gap’ between the lower sums and upper sums. Thus, another equivalent formulation is the Darboux criterion: Theorem 18 T:Darbouxcrit A function f : [a, b] → R is Riemann integrable if and only if for any ε > 0 there is a partition X of [a, b] for which U( f , X) − L( f , X) < ε (3.27) This is an extremely useful result: virtually all of our results on integrability will use the Darboux criterion. Proof. Suppose f is integrable. Then there is a unique real number I which lies between all upper sums and all lower sums. Take any ε > 0. Consider the interval [I, I + ε/2) All the lower sums lie to the left, i.e. ≤ I. All upper sums lie to the right (≥) of I. So since I is the only real number lying between the upper sums and lower sums, there must be at least one upper sum which lies in [I, I + ε/2). Thus, there is a partition Y for which I ≤ U( f ,Y ) < I + ε/2 Similarly, there is a partition Z for which I − ε/2 < L( f , Z) ≤ I. Let X be the partition obtained by pooling together Y and Z. Then I − ε/2 < L( f , Z) ≤ L( f , X) ≤ U( f , X) ≤ U( f , Z) < I + ε/2 (3.28) 66 Ambar N. Sengupta Since both L( f , X) and U( f , X) lie in (I − ε/2, I + ε/2) it follows that U( f , X) − L( f , X) < ε. (3.29) Conversely, suppose that for every ε > 0 there is a partition for which (3.29) holds. Suppose that I and I 0 are distinct real numbers which both lie between all upper sums and all lower sums. Let ε = |I 0 − I| We know that there is a partition X satifying (3.29). Now I and I 0 both lie between U( f , X) and L( f , X). So the difference between I and I 0 is < ε: |I 0 − I| < ε But this contradicts the deifnition of ε taken above. Thus I and I 0 must be equal. So there is a unique real number between all the upper sums and all the lower sums. Thus, f is integrable. QED 3.6 Integrable functions are bounded Suppose f : [a, b] → R is integrable. We will prove that f must be bounded. Theorem 19 Every Riemann integrable function is bounded. Proof. Consider f : [a, b] → R where [a, b] ⊂ R and a < b. Assume that f ∈ R [a, b]. Let us suppose that f is unbounded. Then we will reach a