Los costes de la generación de empleo por parte del turismo

Suppose we associate with each natural numbern a real number a_n. The set of all these numbersa_n, arranged according to the indexn, is called a sequence. We denote this sequence by

{an}^∞_n=1 Thus, the symbol{an}^∞_n=1represents the sequence

a₁, a₂, a₃, . . . , a_n, . . .

For example, the members ofN themselves constitute a sequence when assigned their usual order

1, 2, 3, . . . , n, . . .

This sequence would be denoted by{n}^∞_n=1(becausea_n= n for each n).

Or we could order the elements ofN in a different manner to obtain the sequence 2, 1, 4, 3, 6, 5, 8, 7, . . .

This is quite a different sequence from the sequence{n}^∞_n=1, since the ordering in which the members of the sequence appear is important. Or, if we allow repetitions we get a completely new sequence

1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 6, 7, 8, 8, . . .

(There does not need to be a nice rule involved: it may be impossible to find a formula to describeanin terms ofn.)

Again, we can have a constant sequence

{π}^∞_n=1= π, π, π, π, π, . . . , π, . . . or an alternating (in sign) sequence

{(−1)ⁿ⁺¹}^∞_n=1= +1, −1, +1, −1, +1, −1, . . .

In short, there is no restriction on what the members of a sequence{an}^∞_n=1may be, except that they be real numbers.

Now, some sequences have a rather special property. As we go along the sequence, the numbers in the sequence get arbitrarily closer and closer to some fixed number;

for instance the members of the sequence

1

get arbitrarily closer and closer to 0 asn gets larger, and the members of the sequence



get arbitrarily closer and closer to 1. Again, the members of the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926, . . .

get arbitrarily closer and closer toπ , although this example is not as good as some of the others, since we have not given a general rule for thenth term in the sequence.

If the members of the sequence{an}^∞_n=1get arbitrarily closer and closer to some fixed numbera in this manner, we say that the sequence{an}^∞_n=1tends to the limita, and write

a_n→ a as n → ∞

So far, this is all very intuitive. Let us see if we can obtain a precise definition of what it means to write “an→ a as n → ∞”.

Well, to say thatangets arbitrarily closer and closer toa is to say that the differ-ence|an− a| gets arbitrarily closer and closer to 0. This is the same as saying that whenever is a positive real number, the difference|an− a| is eventually less than

. This leads to the following formal definition:

a_n→ a as n → ∞ iff (∀ > 0)(∃n ∈ N )(∀m ≥ n)(|am− a| < ) This looks quite complicated. Let us try to analyze it. Consider the part

(∃n ∈ N )(∀m ≥ n)(|am− a| < )

This says that there is ann such that for all m greater than or equal to n, the distance froma_m toa is less than . In other words, there is an n such that all terms in the sequence{an}^∞_n=1beyonda_nlie within the distance of a. We can express this concisely by saying that the terms in the sequence{an}^∞_n=1are eventually all within the distance from a.

Thus, the statement

(∀ > 0)(∃n ∈ N )(∀m ≥ n)(|am− a| < )

says that for every > 0, the members of the sequence{an}^∞_n=1are eventually all within the distance from a. This is the formal definition of the intuitive notion of

“a_ngets arbitrarily closer and closer toa”.

Let us consider a numerical example. Consider the sequence{1/n}^∞_n=1. On an intu-itive level, we know that 1/n→ 0 as n → ∞. We shall see how the formal definition works for this sequence. We must prove that

(∀ > 0)(∃n ∈ N )(∀m ≥ n)

1 m− 0

 <  This simplifies at once to

(∀ > 0)(∃n ∈ N )(∀m ≥ n)

1 m < 



To prove that this is a true assertion, let > 0 be arbitrary. We must find an n such that

m≥ n →1 m< 

Pickn large enough so that n > 1/. (This uses the Archimedean property ofR discussed in Exercises 5.8.) If nowm≥ n then

1 m ≤1

n< 

In other words,

One point to notice here is that our choice ofn depended upon the value of . The smaller is, the greater our n needs to be.

Another example is the sequence{n/(n + 1)}^∞_n=1, i.e.,

(1) Formulate both in symbols and in words what it means to say thatan → a as n→ ∞.

(2) Prove that(n/(n+ 1))²→ 1 as n → ∞.

(3) Prove that 1/n²→ 0 as n → ∞.

(4) Prove that 1/2ⁿ→ 0 as n → ∞.

(5) We say a sequence{an}^∞_n=1tends to infinity if, asn increases, a_n increases without bound. For instance, the sequence{n}^∞_n=1tends to infinity, as does the sequence{2ⁿ}^∞_n=1. Formulate a precise definition of this notion, and prove that both of these examples fulfill the definition.

(6) Let{an}^∞_n=1be an increasing sequence (i.e.,a_n< a_n+1for eachn). Suppose thatan→ a as n → ∞. Prove that a = lub{a1, a₂, a₃, . . .}^∞_n=1.

(7) Prove that if{an}^∞_n=1is increasing and bounded above, then it tends to a limit.

This is where you would have found the answers to the exercises — if I had included any. Here is why I have not.

I am sure that, hitherto, all of your math textbooks have had answers in the back of the book. Many current transition books that compete with mine also provide the student with a selection of answers to the exercises. And over the 20 years my book has been in print, many students, some instructors, and even the occasional editor, have suggested that I too should include such a section. On each occasion, after thinking about it at some length (and yes, I know all the arguments in favor of an answer section), I decided against it. I have two reasons for doing this.

First, the book is not written to be read in isolation. It is written for university students who are taking a course that makes the transition from calculus to modern pure mathematics — often known as a “bridge course.” Attempting to answer the problems I include is intended to assist in that process. Knowing whether the answer to a particular problem is correct, however, is of relatively minor importance. Indeed, it may be misleading to know that a particular answer is “correct.” What is at issue with this material is deep understanding. To achieve that, you will inevitably have to discuss any difficulties with your instructor. I want to force you to do just that.

My second reason is this. After completing this book (and the course it accom-panies), you are supposed to be ready to enter the world of real, present-day, pure mathematics — prepared to start working in that discipline. But for the working mathematician, there is no answer in the back of the book — indeed, “the book”

may not yet have been written. Few advanced-level books provide answers in the back (or colored highlighting of important points for that matter, something else I have consistently avoided). To my mind, the sooner you learn to live with that fact, the better (for you). I want you to emerge from working through this book reason-ably well prepared to learn and do some interesting mathematics. Better then that the process of weaning you off from the prepackaged material of high school and first-year university mathematics courses takes place now rather than later, when the going will be considerably tougher. (Although as you are working through this book you will probably feel it can hardly get any tougher!)

Of course, in addition to approaching your instructor, there is nothing to prevent you seeking ideas, and even answers, in other books, or by asking a colleague (in your case a fellow student). Professional mathematicians do that all the time. So should anyone working through this book. But then it will be you who has found the result; you won’t have been given it on a plate by me.

137

∧ , 14

F,  F , 78 (a, b) , 79 A× B , 79

ⁿ , 84 ℵ0,ℵ1 , 84 f : A → B , 87 f (a) , 87 IA , 90 χ_A , 90

f[X] , 98 f◦ g , 100 f⁻¹ , 102 aRb , 114 a−→ b , 122 [a] , 123 {an}^∞_n=1 , 133 an→ a , 134

absolute value, 50, 69, 89

function equality, 92

reflexive, 115, 119 relation, 113, 118 remainder, 50 Russell’s Paradox, 85 sequence, 133 set, 57

set difference, 67 set equality, 58 statement, 11 subset, 59 sufficient, 23 surjection, 96 surjective, 96 symmetric, 116, 119 tend to a limit, 134 tend to infinity, 136 there exists, 29 transformation, 98 transitive, 116, 119 triangle inequality, 71

triple, 80 truth, 54 truth table, 24 tuple, 80 uncountable, 108 underlying set, 114 union, 61, 75 unique existence, 39 universal quantifier, 31 universal set, 61 unless, 27 unordered pair, 79 upper bound, 131 value, 88 Venn diagram, 62 vertex, 121 whenever, 23

Zermelo–Fraenkel set theory, 85