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Page 018

the analysis of the real numbers, sequences and series of real numbers, and real-valued functions. This is related to, but is distinct from, complex analysis, which concerns the analysis of the complex numbers and complex functions, harmonic analysis, which concerns the analysis of harmonics (waves) such as sine waves, and how they synthesize other functions via the Fourier transform, functional analysis, which focuses much more heavily on functions (and how they form things like vector spaces), and so forth.

Page 019

There is a certain philosophical satisfaction in knowing why things work, but a pragmatic person may argue that one only needs to know how things work to do real-life problems.

Page 019

you can certainly use things like the chain rule, L’Hˆopital’s rule, or integration by parts without knowing why these rules work, or whether there are any exceptions to these rules. However, one can get into trouble if

Page 020

one applies rules without knowing where they came from and what the limits of their applicability are

Page 028

The analysis you learn in this text will help you resolve these questions, and will let you know when these rules (and others) are justified, and when they are illegal, thus separating the useful applications of these rules from the nonsense. Thus they can prevent you from making mistakes, and can help you place these rules in a wider context. Moreover, as you learn analysis you will develop an “analytical way of thinking”, which will help you whenever you come into contact with any new rules of mathematics, or when dealing with situations which are not quite covered by the standard rules,

Page 029

You will develop a sense of why a rule in mathematics (e.g., the chain rule) works, how to adapt it to new situations, and what its limitations (if any) are; this will allow you to apply the mathematics you have already learnt more confidently and correctly

Page 030

This is not an arbitrary choice of rule; it can be proven from more primitive, and more fundamental, properties of the number system.

Page 031

We will try to introduce these concepts one at a time and identify explicitly what our assumptions are as we go along - and not allow ourselves to use more “advanced” tricks such as the rules of algebra until we have actually proven them. This may seem like an irritating constraint, especially as we will spend a lot of time proving statements which are “obvious”, but it is necessary to do this suspension of known facts to avoid circularity

Page 031

the results here may seem trivial, but the journey is much more important than the destination, for now.

Page 033

we can define complicated operations in terms of simpler operations.

Page 033

incrementing seems to be a fundamental operation, not reducible to any simpler operation

Page 034

It may seem that this is enough to describe the natural numbers. However, we have not pinned down completely the behavior of N even with our new axiom, it is still possible that our number system behaves in other pathological ways

Page 036

Because this axiom refers not just to variables, but also properties, it is of a different nature than the other four axioms;

Page 037

technically be called an axiom schema rather than an axiom - it is a template for producing an (infinite) number of axioms

Page 038

A remarkable accomplishment of modern analysis is that just by starting from these five very primitive axioms, and some additional axioms from set theory, we can build all the other number systems, create functions, and do all the algebra and calculus that we are used to.

Page 039

Note that our definition of the natural numbers is axiomatic rather than constructive. We have not told you what the natural numbers are (so we do not address such questions as what the numbers are made of, are they physical objects, what do they measure, etc.) - we have only listed some things you can do with them (in fact, the only operation we have defined on them right now is the increment one) and some of the properties that they have. This is how mathematics works - it treats its objects abstractly, caring only about what properties the objects have, not what the objects are or what they mean.

Page 039

there are multiple ways to construct a working model of the natural numbers, and it is pointless, at least from a mathematician’s standpoint, as to argue about which model is the “true” one

Page 039

Historically, the realization that numbers could be treated axiomatically is very recent, not much more than a hundred years old. Before then, numbers were generally understood to be inextricably connected to some external concept, such as counting the cardinality of a set, measuring the length of a line segment, or the mass of a physical object, etc. This worked reasonably well, until one was forced to move from one number system to another;

Page 039

thus each great advance in the theory of numbers - negative numbers, irrational numbers, complex numbers, even the number zero - led to a lot of unnecessary philosophical anguish

Page 040

a mathematician can use any of these models when it is convenient, to aid his or her intuition and understanding, but they can also be just as easily discarded when they begin to get in the way.

Page 040

By using all the axioms together we will now conclude that this procedure will give a single value to the sequence element an for each natural number n. More precisely3:

Page 040

Proposition 2.1.16 (Recursive definitions)

Page 044

Once we have a notion of addition, we can begin defining a notion of order.

Page 045

The properties of order allow one to obtain a stronger version of the principle of induction

Page 048

Now that we have the familiar operations of addition and multiplication, the more primitive notion of increment will begin to fall by the wayside, and we will see it rarely from now on.

Page 051

There is a special case of set theory, called “pure set theory”, in which all objects are sets

Page 051

From a logical point of view, pure set theory is a simpler theory, since one only has to deal with sets and not with objects; however, from a conceptual point of view it is often easier to deal with impure set theories in which some objects are not considered to be sets. The two types of theories are more or less equivalent for the purposes of doing mathematics, and so we shall take an agnostic position as to whether all objects are sets or not.

Page 051

If A is not a set, we leave the statement x ∈ A undefined; for instance, we consider the statement 3 ∈ 4 to neither be true or false, but simply meaningless, since 4 is not a set

Page 057

Axiom 3.5 (Axiom of specification). Let A be a set, and for each x ∈ A, let P(x) be a property pertaining to x (i.e., P(x) is either a true statement or a false statement). Then there exists a set, called {x ∈ A : P(x) is true} (or simply {x ∈ A : P(x)} for short), whose elements are precisely the elements x in A for which P(x) is true. In other words, for any object y, y ∈ {x ∈ A : P(x) is true} ⇐⇒ (y ∈ A and P(y) is true). This axiom is also known as the axiom of separation.

Page 058

Remark 3.1.26. By the way, one should be careful with the English word “and”: rather confusingly, it can mean either union or intersection, depending on context. For instance, if one talks about a set of “boys and girls”, one means the union of a set of boys with a set of girls, but if one talks about the set of people who are single and male, then one means the intersection of the set of single people with the set of male people. (Can you work out the rule of grammar that determines when “and” means union and when “and” means intersection?)

Page 064

The problem with the above axiom is that it creates sets which are far too “large” - for instance, we can use that axiom to talk about the set of all objects (a so-called “universal set”). Since sets are themselves objects (Axiom 3.1), this means that sets are allowed to contain themselves, which is a somewhat silly state of affairs.

Page 065

Exercise 3.2.1. Show that the universal specification axiom, Axiom 3.8, if assumed to be true, would imply Axioms 3.2, 3.3, 3.4, 3.5, and 3.6. (If we assume that all natural numbers are objects, we also obtain Axiom 3.7.) Thus, this axiom, if permitted, would simplify the foundations of set theory tremendously (and can be viewed as one basis for an intuitive model of set theory known as “naive set theory”). Unfortunately, as we have seen, Axiom 3.8 is “too good to be true”!

Page 065

if Axiom 3.8 is true, then a universal set exists, and conversely, if a universal set exists, then Axiom 3.8 is true.

Page 071

The concepts of injectivity and surjectivity are in many ways dual to each other

Page 072

If f is bijective, then for every y ∈ Y , there is exactly one x such that f(x) = y (there is at least one because of surjectivity, and at most one because of injectivity).

Page 077

Zermelo-Fraenkel-Choice (ZFC ) axioms of set theory

Page 081

Also, the empty Cartesian product 1≤i≤0 Xi gives, not the empty set {}, but rather the singleton set {()} whose only element is the 0-tuple (), also known as the empty tuple.

Page 084

Exercise 3.5.13. The purpose of this exercise is to show that there is essentially only one version of the natural number system in set theory

Page 085

the Peano axiom approach treats natural numbers more like ordinals than cardinals. (The cardinals are One, Two, Three, ..., and are used to count how many things there are in a set. The ordinals are First, Second, Third, ..., and are used to order a sequence of objects. There is a subtle difference between the two, especially when comparing infinite cardinals with infinite ordinals, but this is beyond the scope of this text).

Page 091

this definition is circular because it requires a notion of subtraction, which we can only adequately define once the integers are constructed

Page 091

We still have to resolve (c). To get around this problem we will use the following work-around: we will temporarily write integers not as a difference a − b, but instead use a new notation a−−b to define integers, where the −− is a meaningless place-holder

Page 106

It is often useful to think of the notion of “ε-close” as an approximate substitute for that of equality in analysis.

Page 111

You are of course free to design your own number system, possibly including one where division by zero is permitted; but you will have to give up one or more of the field axioms from Proposition 4.2.4, among other things, and you will probably get a less useful number system in which to do any real-world problems.

Page 112

The procedure we give here of obtaining the real numbers as the limit of sequences of rational numbers may seem rather complicated. However, it is in fact an instance of a very general and useful procedure, that of completing one metric space to form another

Page 131

while positive real numbers can be arbitrarily large or small, they cannot be larger than all of the positive integers, or smaller in magnitude than all of the positive rationals

Page 131 硬件的極限：

Proposition 5.4.12 (Bounding of reals by rationals). Let x be a positive real number. Then there exists a positive rational number q such that q ≤ x, and there exists a positive integer N such that x ≤ N

Page 139

Proposition 5.6.3. All the properties in Propositions 4.3.10 and 4.3.12 remain valid if x and y are assumed to be real numbers instead of rational numbers. Instead of giving an actual proof of this proposition, we shall give a meta-proof (an argument appealing to the nature of proofs, rather than the nature of real and rational numbers)

Page 146

We sometimes use the phrase “an → x as n → ∞” as an alternate way of writing the statement “(an)∞ n=m converges to x”. Bear in mind, though, that the individual statements an → x and n → ∞ do not have any rigorous meaning; this phrase is just a convention

Page 146

The notation limn→∞ an does not give any indication about the starting index m of the sequence, but the starting index is irrelevant

Page 147

Convergent sequences are Cauchy

Page 152

To avoid these issues we shall simply not define any arithmetic operations on the extended real number system other than negation and order.

Page 158

Limits are of course a special case of limit points

Page 161

Let c be a real number. If (an)∞ n=m converges to c, then we must have L+ = L− = c. Conversely, if L+ = L− = c, then (an)∞ n=m converges to c.

Page 163

Theorem 6.4.18 (Completeness of the reals). A sequence (an)∞ n=1 of real numbers is a Cauchy sequence if and only if it is convergent.

Page 163

Remark 6.4.19. Note that while this is very similar in spirit to Proposition 6.1.15, it is a bit more general, since Proposition 6.1.15 refers to Cauchy sequences of rationals instead of real numbers

Page 164

doing analysis (taking limits, taking derivatives and integrals, finding zeroes of functions, that kind of thing

Page 164

In the language of metric spaces (see Chapter B.2), Theorem 6.4.18 asserts that the real numbers are a complete metric space - that they do not contain “holes” the same way the rationals do.

Page 164

(Certainly the rationals have lots of Cauchy sequences which do not converge to other rationals; take for instance the sequence 1, 1.4, 1.41, 1.414, 1.4142,... which converges to the irrational √2.)

Page 168

Theorem 6.6.8 (Bolzano-Weierstrass theorem). Let (an)∞ n=0 be a bounded sequence (i.e., there exists a real number M > 0 such that |an| ≤ M for all n ∈ N). Then there is at least one subsequence of (an)∞ n=0 which converges.

Page 168

The Bolzano-Weierstrass theorem says that if a sequence is bounded, then eventually it has no choice but to converge in some places; it has “no room” to spread out and stop itself from acquiring limit points.

Page 168

In the language of topology, this means that the interval {x ∈ R : −M ≤ x ≤ M} is compact, whereas an unbounded set such as the real line R is not compact.

Page 173

The difference between “sum” and “series” is a subtle linguistic one. Strictly speaking, a series is an expression of the form n i=m ai; this series is mathematically (but not semantically) equal to a real number, which is then the sum of that series.

Page 177

often abbreviate x∈{y∈X:P(y) is true} f(x) as x∈X:P(x) is true f(x) or even as P(x) is true f(x)

Page 181

the notation am +am+1 +am+2 +... is of course designed to look very suggestively like a sum, but is not actually a finite sum because of the “...” symbol.

Page 183

This Proposition, by itself, is not very handy, because it is not so easy to compute the partial sums q n=p an in practice. However, it has a number of useful corollaries.

Page 184 A clue for the weird relationship between absolutely convergent and conditionally convergent.

Proposition 7.2.12 (Alternating series test). Let (an)∞ n=m be a sequence of real numbers which are non-negative and decreasing, thus an ≥ 0 and an ≥ an+1 for every n ≥ m. Then the series ∞ n=m(−1)nan is convergent if and only if the sequence an converges to 0 as n → ∞.

Page 187

Corollary 7.3.2 (Comparison test)

Page 187 useful for understanding series

in fact ∞ n=m an ≤ ∞ n=m |an| ≤ ∞ n=m bn.

Page 188 We have the proof, but how actually Cauchy got such an unusual result, by his intuition?

Cauchy criterion

Page 188

Lemma 7.3.3 (Geometric series). Let x be a real number. If |x| ≥ 1, then the series ∞ n=0 xn is divergent. If however |x| < 1, then the series is absolutely convergent and ∞ n=0 xn = 1/(1 − x)

Page 188

Proposition 7.3.4 (Cauchy criterion). Let (an)∞ n=1 be a decreasing sequence of non-negative real numbers (so an ≥ 0 and an+1 ≤ an for all n ≥ 1). Then the series ∞ n=1 an is convergent if and only if the series ∞ k=0 2ka2k = a1 + 2a2 + 4a4 + 8a8 + ... is convergent.

Page 188

An interesting feature of this criterion is that it only uses a small number of elements of the sequence

Page 190

the series ∞ n=1 1/n (also known as the harmonic series) is divergent, as claimed earlier. However, the series ∞ n=1 1/n2 is convergent.

Page 190

The quantity ∞ n=1 1/nq, when it converges, is called ζ(q), the Riemann-zeta function of q.

Page 193

when the series is not absolutely convergent, then the rearrangements are very badly behaved.

Page 194 It means the criterion whether a series can be rearranged is not clear, or still not found.

This is not to say that rearranging a series that is not absolutely convergent necessarily gives you the wrong answer

Page 194

This is not to say that rearranging a series that is not absolutely convergent necessarily gives you the wrong answer - for instance, in theoretical physics one often performs similar maneuvres, and one still (usually) obtains a correct answer at the end - but doing so is risky

Page 197

Corollary 7.5.3 (Ratio test). Let ∞ n=m an be a series of nonzero numbers. (The non-zero hypothesis is required so that the ratios |an+1|/|an| appearing below are well-defined.) • If lim supn→∞ |an+1| |an| < 1, then the series ∞ n=m an is absolutely convergent (hence conditionally convergent). • If lim infn→∞ |an+1| |an| > 1, then the series ∞ n=m an is not conditionally convergent (and thus cannot be absolutely convergent). • In the remaining cases, we cannot assert any conclusion.

Page 198

A set X is said to be countably infinite (or just countable) iff it has equal cardinality with the natural numbers N. A set X is said to be at most countable iff it is either countable or finite.

Page 199

a countable set can be arranged in a sequence

Page 199

countable sets are infinite

Page 204

The rationals Q are countable.

Page 204

it is quite difficult (though not impossible) to actually try and come up with an explicit sequence a0, a1,... for Q.

Page 210

Lemma 8.2.7. Let ∞ n=0 an be a series of real numbers which is conditionally convergent, but not absolutely convergent. Define the sets A+ := {n ∈ N : an ≥ 0} and A− := {n ∈ N : an < 0}, thus A+ ∪ A− = N and A+ ∩ A− = ∅. Then both of the series n∈A+ an and n∈A− an are not conditionally convergent (and thus not absolutely convergent).

Page 210

a series which converges conditionally but not absolutely can be rearranged to converge to any value

Page 212

(Cantor’s theorem). Let X be an arbitrary set (finite or infinite). Then the sets X and 2X cannot have equal cardinality.

Page 214

One could ask whether there exist any sets which have strictly larger cardinality than the natural numbers, but strictly smaller cardinality than the reals. The Continuum Hypothesis asserts that no such sets exist. Interestingly, it was shown in separate works of Kurt G¨odel (1906– 1978) and Paul Cohen (1934–2007) that this hypothesis is independent of the other axioms of set theory;

Page 215 Then can we build another axiom system so that we can prove it?

it can neither be proved nor disproved in that set of axioms (unless those axioms are inconsistent, which is highly unlikely).

Page 216

One reason for this confidence is a theorem due to the great logician Kurt G¨odel, who showed that a result proven using the axiom of choice will never contradict a result proven without the axiom of choice (unless all the other axioms of set theory are themselves inconsistent, which is highly unlikely)

Page 216

More precisely, G¨odel demonstrated that the axiom of choice is undecidable; it can neither be proved nor disproved from the other axioms of set theory, so long as those axioms are themselves consistent.

Page 216

“real-life” application of analysis (more precisely, any application involving only “decidable” questions)

Page 217

Axiom 8.1 (Choice). Let I be a set, and for each α ∈ I, let Xα be a non-empty set. Then α∈I Xα is also non-empty. In other words, there exists a function (xα)α∈I which assigns to each α ∈ I an element xα ∈ Xα.

Page 217

the axiom of choice can lead to proofs which are non-constructive - demonstrating existence of an object without actually constructing the object explicitly

Page 219

Strictly speaking, a partially ordered set is not a set X, but rather a pair (X, ≤X)

Page 223

Lemma 8.5.15 (Zorn’s lemma). Let X be a non-empty partially ordered set, with the property that every totally ordered subset Y of X has an upper bound. Then X contains at least one maximal element.

Page 224

Zorn’s lemma (also called the principle of transfinite induction)

Page 226 In real world infinity may not exisit, so the theory concerning infinity is over-simulating, but we can't (easily/exactly) count the amount, so we use infinity as a tool for suggesting a virtual upper bound.

logically equivalent to the axiom of choice

Page 226

well ordering principle: every set X has at least one well-ordering.

Page 226

if Hausdorff’s maximality principle is true, then Zorn’s lemma is true. Tthus by Exercise 8.5.17,

Page 226

We thus see that the axiom of choice, Zorn’s lemma, and the well-ordering principle are all logically equivalent to each other.

Page 226

the cardinality of any two sets is comparable, as long as one assumes the axiom of choice.

Page 229

the real line R itself is the open interval (−∞, +∞), while the extended real line R∗ is the closed interval [−∞, +∞].

Page 230

The closure of X, sometimes denoted X is defined to be the set of all the adherent points of X.

Page 230

Let X be a subset of R, and let x ∈ R. We say that x is an adherent point of X iff it is ε-adherent to X for every ε > 0.

Page 231

The closure of Q is R

Page 232

N, Z, R, ∅ are closed, while Q is not.

Page 232

the set of adherent points splits into the set of limit points and the set of isolated points

Page 233

Heine-Borel theorem for the line

Page 233

This theorem shall play a key rˆole in subsequent sections of this chapter. In the language of metric space topology, it asserts that every subset of the real line which is closed and bounded, is also compact

Page 236

One can certainly study a function through its graph, by using the geometry of the plane R2 (e.g., employing such concepts as tangent lines, area, and so forth). We however will pursue a more “analytic” approach, in which we rely instead on the properties of the real numbers to analyze these functions. The two approaches are complementary; the geometric approach offers more visual intuition, while the analytic approach offers rigour and precision.

Page 261

The difference between uniform continuity and continuity is that in uniform continuity one can take a single δ which works for all x0 ∈ X; for ordinary continuity, each x0 ∈ X might use a different δ.

Page 264

if the domain of the function is a closed interval, then continuous functions are in fact uniformly continuous

Page 268

these definitions can be modified to handle functions of several variables, or functions whose values are vectors instead of scalar. Furthermore, one’s geometric intuition becomes difficult to rely on once one has more than three dimensions in play. (Conversely, one can use one’s experience in analytic rigour to extend one’s geometric intuition to such abstract settings; as mentioned earlier, the two viewpoints complement rather than oppose each other.)

Page 268

f is differentiable at x0 on X with derivative L

Page 270

we will refrain from using the notation df dx whenever it could possibly lead to confusion. (This confusion becomes even worse in the calculus of several variables, and the standard notation of ∂f ∂x can lead to some serious ambiguities

Page 273

If one writes y for f(x), and z for g(y), then the chain rule can be written in the more visually appealing manner dz dx = dz dy dy dx . However, this notation can be misleading (for instance it blurs the distinction between dependent variable and independent variable, especially for y), and leads one to believe that the quantities dz, dy, dx can be manipulated like real numbers. However, these quantities are not real numbers (in fact, we have not assigned any meaning to them at all)

Page 273

It is possible to think of dy, dx, etc. as “infinitesimal real numbers” if one knows what one is doing, but for those just starting out in analysis, I would not recommend this approach, especially if one wishes to work rigorously. (There is a way to make all of this rigorous, even for the calculus of several variables, but it requires the notion of a tangent vector, and the derivative map, both of which are beyond the scope of this text.)

Page 275

Local extrema are stationary

Page 295

Compare this definition to the relationship between the lim sup, lim inf, and limit of a sequence an that was established in Proposition 6.4.12(f); the lim sup is always greater than or equal to the lim inf, but they are only equal when the sequence converges, and in this case they are both equal to the limit of the sequence. The definition given above may differ from the definition you may have encountered in your calculus courses, based on Riemann sums. However, the two definitions turn out to be equivalent

Page 296

The restriction J = ∅ is required because the quantities infx∈J f(x) and supx∈J f(x) are infinite (or negative infinite) if J is empty.

Page 305

This gives a large class of Riemann integrable functions already; the bounded continuous functions. But we can expand this class a little more, to include the bounded piecewise continuous functions.

Page 312

We now have enough machinery to connect integration and differentiation via the familiar fundamental theorem of calculus

Page 319

the advantage of the Riemann-Stieltjes integral is that it still makes sense even when α is not differentiable.

Page 322

The big advantage of writing logically, however, is that one can be absolutely sure that your conclusion will be correct, as long as all your hypotheses were correct and your steps were logical; using other styles of writing one can be reasonably convinced that something is true, but there is a difference between being convinced and being sure.

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