Notes in Introductory Real Analysis, Exams of Calculus

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Notes in Introductory Real Analysis

Ambar N. Sengupta

March, 2014

2 Ambar N. Sengupta

• Introductory Remarks
• 1 Ordered Fields and The Real Number System
  • 1.1 Ordered Fields
    • 1.1.1 Fields
    • 1.1.2 Order Relations
    • 1.1.3 Ordered Fields
    • 1.1.4 The absolute value function
    • 1.1.5 The Archimedean Property
  • 1.2 The Real Number System R
    • 1.2.1 Hilbert Maximality and the Completeness Property
    • 1.2.2 Completeness of R and measurement
    • Problem Set
• 2 The Extended Real Line and Its Topology
  • 2.1 The extended real line
  • 2.2 Neighborhoods
  • 2.3 Types of points for a set
  • 2.4 Interior, Exterior, and Boundary of a Set
  • 2.5 Open Sets and Topology
  • 2.6 Closed Sets
  • 2.7 Open Sets and Closed Sets
  • 2.8 Closed sets in R and in R∗
  • 2.9 Closure of a set
  • 2.10 The closure of a set is closed
  • 2.11 S is the smallest closed set containing S
  • 2.12 R∗ is compact
  • 2.13 Compactness of closed subsets of R∗.
  • 2.14 The Heine-Borel Theorem: Compact subsets of R
  • 2.15 Sequences 4 Ambar N. Sengupta
  • 2.16 Limits points of a sequence
  • 2.17 Bolzano-Weierstrass Theorem
  • 2.18 Limit points and suprema and infima
  • 2.19 Limit of a sequence
  • 2.20 Simple Examples of limits
  • 2.21 The sequence 1/n
  • 2.22 The sequence Rn
  • 2.23 Monotone Sequences
  • 2.24 The limit of a sequence is unique
  • 2.25 Convergent sequences and Cauchy sequences
  • 2.26 Every Cauchy sequence is bounded
  • 2.27 Every Cauchy sequence is convergent
  • 2.28 The rationals are countable
  • 2.29 The real numbers are uncountable
  • 2.30 Connected Sets
• 3 Integration
  • 3.1 Approaching the Riemann Integral
  • 3.2 Riemann Sums
  • 3.3 Definition of the Riemann Integral
  • 3.4 Refining Partitions
  • 3.5 The Darboux Criterion
  • 3.6 Integrable functions are bounded
  • 3.7 Variation of a Function
  • 3.8 The algebra R [a, b]
  • 3.9 C[a, b] ⊂ R [a, b]
  • 3.10 The Integral as a Non-negative Linear Functional
  • 3.11 Additivity of the Integral
  • 3.12 Monotone Functions are Riemann Integrable
  • 3.13 Riemann Sums and the Riemann Integral
    • Bibliography
• A Question Bank

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 [5]. 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).

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 ZW

PQ

RS

if they are unequal and if whenever mZW>nXY then also mRS>nPQ.

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 n

p 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 [3]). 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 [1], 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 [4] 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.

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

= b−^1 ,

and the product ab−^1 as a b

= ab−^1

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

1 a = a,

where now 1 is the number one in Z. Next,

2 a def = a + a,

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 x^2 > 0

(iii) 1 > 0

(iv) For any r ∈ Z the element r 1 ∈ 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

16 Ambar N. Sengupta

Going further, if m, n ∈ Z, and n 6 = 0, we have then the ratio

m n

∈ F.

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| =

x if x ≥ 0 −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

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 [2].

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.