lecture notes in economics and mathematical systems and mathematics for economists an introductory textbook pdf free download
Lecture Notes on Mathematics for Economists
First Version: August 9, 2007
This Version: May 18, 2010
Summer 2010, Department of Economics, McGill University
August 16 - 27 (tentative): Monday - Friday, 10:00am - 1:00pm; at TBA
Instructor: Takashi Kunimoto
Class Web: http://people.mcgill.ca/takashi.kunimoto/?View=Publications
Oﬃce: Leacock 438
COURSE DESCRIPTION: This course is designed to provide you with
mathematical tools that are extensively used in graduate economics courses. The topics
which will be covered is
• Sets and Functions;
• Topology in the Euclidean Space;
• Linear Algebra
• Multivariate Calculus;
• Static Optimization;
• (Optional) Correspondences and Fixed Points; and
• (Optional) The First-Order Diﬀerential Equations in One Variable.
A good comprehension of the material covered in the notes is essential for successful
graduate studies in economics. Since we are seriously time constrained – which you
might not believe –, it would be very useful for you to carry one of the books provided
below as a reference after you start to study in graduate school in September.
The main textbook of this course is “Further Mathematics for Economic Analysis.” I
mostly use this book for the course. However, if you don’t ﬁnd the main textbook helpful
enough, I strongly recommend you that you should buy at least one of the other books
I listed below as well as “Further Mathematics for Economic Analysis.” Of course, you
can buy any math book which you ﬁnd useful.
I am thankful to the students for their comments, questions, and suggestions. Yet, I believe that
there are still many errors in this manuscript. Of course, all remaining ones are my own.
1• “Further Mathematics for Economic Analysis,” by Knut Sydsaeter, Peter Ham-
mond, Atle Seierstad, and Atle Strom, Prentice Hall, 2005 (Main Textbook.If
you don’t have any math book or are not conﬁdent about your math skill, this
book will help you a lot.)
• “Mathematical Appendix,” in Advanced Microeconomic Theory, Second Edition,
by Geoﬀrey A. Jehle and Philip J. Reny, (2000), Addison Wesley (Supplementary.
This is the main textbook for Econ 610 but the mathematical appendix of this
book is too concise in many times)
• “Mathematics for Economists,” by Simon and Blume, Norton, (1994). (Supple-
mentary. This book is a popular math book in many Ph.D programs in economics.
There has to be a reason for that, although I don’t know the true one.)
• “Fundamental Methods of Mathematical Economics,” by A. Chiang, McGraw-Hill.
(more elementary and supplementary)
• “Introductory Real Analysis,” by A. N. Kolmogorov and S.V. Fomin, Dover Publi-
cations (very very advanced and supplementary. If you really like math, this is the
book for you.)
OFFICE HOURS: Wednesday and Friday, 2:00pm - 3:00pm
PROBLEM SETS: There will be several problem sets. Problem sets are essential to
help you understand the course and to develop your skill to analyze economic problems.
ASSESSMENT: No grade will be assigned. However, you are expected to do the
problem sets assigned.
1 Introduction 6
2 Preliminaries 9
2.1 Logic . . . .... ... .... ... ... .... ... .... ... ... .. 9
2.1.1 Necessity and Suﬃciency . . . . .... ... .... ... ... .. 9
2.1.2 Theorems and Proofs . . . . . . .... ... .... ... ... .. 10
2.2 Set Theory . . . . . . .... ... ... .... ... .... ... ... .. 11
2.3 Relations .... ... .... ... ... .... ... .... ... ... .. 13
2.3.1 Preference Relations . . . . . . . .... ... .... ... ... .. 13
2.4 Functions .... ... .... ... ... .... ... .... ... ... .. 14
2.4.1 Least Upper Bound Principle . . .... ... .... ... ... .. 15
3 Topology in R 17
3.1 Sequences on R . ... .... ... ... .... ... .... ... ... .. 17
3.1.1 Subsequences . .... ... ... .... ... .... ... ... .. 18
3.1.2 Cauchy Sequences . . . . . . . . .... ... .... ... ... .. 19
3.1.3 Upper and Lower Limits . . . . . .... ... .... ... ... .. 20
3.1.4 Inﬁmum and Supremum of Functions . . . . .... ... ... .. 21
3.1.5 Indexed Sets . .... ... ... .... ... .... ... ... .. 21
3.2 Point Set Topology in R .. ... ... .... ... .... ... ... .. 21
3.3 Topology and Convergence . . . . . . . .... ... .... ... ... .. 23
3.4 Properties of Sequences in R ... ... .... ... .... ... ... .. 24
3.5 Continuous Functions .... ... ... .... ... .... ... ... .. 26
4 Linear Algebra 29
4.1 Basic Concepts in Linear Algebra . . . . .... ... .... ... ... .. 29
4.2 Determinants and Matrix Inverses . . . .... ... .... ... ... .. 32
4.2.1 Determinants . .... ... ... .... ... .... ... ... .. 32
4.2.2 Matrix Inverses .... ... ... .... ... .... ... ... .. 32
4.2.3 Cramer’s Rule .... ... ... .... ... .... ... ... .. 33
4.3 Vectors . .... ... .... ... ... .... ... .... ... ... .. 34
4.4 Linear Independence . .... ... ... .... ... .... ... ... .. 35
4.4.1 Linear Dependence and Systems of Linear Equations . . . . . . . . 36
4.5 Eigenvalues . . . . . . .... ... ... .... ... .... ... ... .. 37
34.5.1 Motivations . . .... ... ... .... ... .... ... ... .. 37
4.5.2 How to Find Eigenvalues . . . . .... ... .... ... ... .. 39
4.6 Diagonalization . . . . .... ... ... .... ... .... ... ... .. 40
4.7 Quadratic Forms . . . .... ... ... .... ... .... ... ... .. 42
4.8 Appendix 1: Farkas Lemma . . . . . . . .... ... .... ... ... .. 44
4.8.1 Preliminaries . .... ... ... .... ... .... ... ... .. 45
4.8.2 Fundamental Theorem of Linear Algebra . . .... ... ... .. 46
4.8.3 Linear Inequalities . . . . . . . . .... ... .... ... ... .. 47
4.8.4 Non-Negative Solutions . . . . . .... ... .... ... ... .. 47
4.8.5 The General Case . . . . . . . . .... ... .... ... ... .. 49
4.9 Appendix 2: Linear Spaces . . . . . . . .... ... .... ... ... .. 49
4.9.1 Number Fields .... ... ... .... ... .... ... ... .. 49
4.9.2 Deﬁnitions . . .... ... ... .... ... .... ... ... .. 51
4.9.3 Bases, Components, Dimension . .... ... .... ... ... .. 52
4.9.4 Subspaces . . . .... ... ... .... ... .... ... ... .. 53
4.9.5 Morphisms of Linear Spaces . . . .... ... .... ... ... .. 54
5 Calculus 55
5.1 Functions of a Single Variable . . . . . . .... ... .... ... ... .. 55
5.2 Real-Valued Functions of Several Variables . . . . . .... ... ... .. 56
5.3 Gradients .... ... .... ... ... .... ... .... ... ... .. 56
5.4 The Directional Derivative . . . . . . . . .... ... .... ... ... .. 57
5.5 Convex Sets . . . . . . .... ... ... .... ... .... ... ... .. 58
5.5.1 Upper Contour Sets . . . . . . . .... ... .... ... ... .. 59
5.6 Concave and Convex Functions . . . . . .... ... .... ... ... .. 59
5.7 Concavity/Convexity for C Functions . .... ... .... ... ... .. 60
5.7.1 Jensen’s Inequality . . . . . . . . .... ... .... ... ... .. 63
5.8 Quasiconcave and Quasiconvex Functions . . . . . . .... ... ... .. 64
5.9 Total Diﬀerentiation . .... ... ... .... ... .... ... ... .. 68
5.9.1 Linear Approximations and Diﬀerentiability . .... ... ... .. 69
5.10 The Inverse of a Transformation . . . . .... ... .... ... ... .. 72
5.11 Implicit Function Theorems . . . . . . . .... ... .... ... ... .. 73
6 Static Optimization 77
6.1 Unconstrained Optimization . . . . . . . .... ... .... ... ... .. 77
6.1.1 Extreme Points .... ... ... .... ... .... ... ... .. 77
6.1.2 Envelope Theorems for Unconstrained Maxima . . . . . . . . . . . 78
6.1.3 Local Extreme Points . . . . . . .... ... .... ... ... .. 79
6.1.4 Necessary Conditions for Local Extreme Points . . . . . . . . . . . 80
6.2 Constrained Optimization . . . . . . . . .... ... .... ... ... .. 81
6.2.1 Equality Constraints: The Lagrange Problem .... ... ... .. 81
6.2.2 Lagrange Multipliers as Shadow Prices . . . . .... ... ... .. 84
6.2.3 Tangent Hyperplane . . . . . . . .... ... .... ... ... .. 84
6.2.4 Local First-Order Necessary Conditions . . . .... ... ... .. 85
46.2.5 Second-Order Necessary and Suﬃcient Conditions for Local Ex-
treme Points . .... ... ... .... ... .... ... ... .. 85
6.2.6 Envelope Result for Lagrange Problems . . . .... ... ... .. 87
6.3 Inequality Constraints: Nonlinear Programming . . . .... ... ... .. 88
6.4 Properties of the Value Function . . . . .... ... .... ... ... .. 90
6.5 Constraint Qualiﬁcations . . . . . . . . .... ... .... ... ... .. 91
6.6 Nonnegativity Constraints . . . . . . . . .... ... .... ... ... .. 94
6.7 Concave Programming Problems . . . . .... ... .... ... ... .. 95
6.8 Quasiconcave Programming . . . . . . . .... ... .... ... ... .. 96
6.9 Appendix: Linear Programming . . . . . .... ... .... ... ... .. 96
7 Diﬀerential Equations 97
7.1 Introduction . . . . . . .... ... ... .... ... .... ... ... .. 97
8 Fixed Point Theorems 98
8.1 Banach Fixed Point Theorem . . . . . . .... ... .... ... ... .. 98
8.2 Brouwer Fixed Point Theorem . . . . . .... ... .... ... ... .. 99
9 Topics on Convex Sets 102
9.1 Separation Theorems . .... ... ... .... ... .... ... ... .. 102
9.2 Polyhedrons and Polytopes . . . . . . . .... ... .... ... ... .. 104
9.3 Dimension of a Set . . .... ... ... .... ... .... ... ... .. 105
9.4 Properties of Convex Sets . . . . . . . . .... ... .... ... ... .. 105
I start my lecture with Rakesh Vohra’s message about what economic theory is. He is a
professor at Northwestern University.
All of economic theorizing reduces, in the end, to the solution of one of
Given a function f and a set S:
1. Find an x such that f(x)is in S. This is the feasibility question.
2. Find an x in S that optimizes f(x). This is the problem of optimality.
3. Find an x in S such that f(x)= x, this is the ﬁxed point problem.
These three problems are, in general, quite diﬃcult. However, if one is
prepared to make assumptions about the nature of the underlying function
(say it is linear, convex or continuous) and the nature of the set S (convex,
compact etc.) it is possible to provide answers and very nice ones at that.
I think this is the biggest picture of economic theory you could have as you go along
this course. Whenever you are at a loss, please come back to this message.
We build our theory on individuals. Assume that all commodities are traded in the
centralized markets. Throughout Econ 610 and 620, we assume that each individual
(consumer and ﬁrm) takes prices as given. We call this the price taking behavior assump-
tion. You might ask why individuals are price takers. My answer would be “why not?”
Let us go as far as we can with this behavioral assumption and thereafter try to see the
limitation of the assumption. However, you have to wait for Econ 611 and 621 for how
to relax this assumption. So, stick with this assumption. For each consumer, we want
1. What is the set of “physically feasible bundles? Is there any such a bundle at all
(feasibility)? We call this set the consumption set.
See Preface of Advanced Mathematical Economics by Rakesh V. Vohra.
62. What is the set of “ﬁnancially feasible bundles? Is there any such a bundle at all
(feasibility)? We call this set the budget set.
3. What is the best bundle to the consumer among all feasible bundles (optimality)?
We call this bundle the consumer’s demand.
We can make the exact parallel argument for the ﬁrm. What is the set of “technically”
feasible inputs (feasibility)? We call this the production set of the ﬁrm. What is the
best combination of inputs to maximize its proﬁt (optimality)? We call this the ﬁrm’s
supply. Once we ﬁgure out what are feasible and best choices to each consumer and each
ﬁrm under any possible circumstance, we want to know if there is any coherent state of
aﬀairs where everybody makes her best choice. In particular, all markets must clear. We
call this coherent state “competitive (Walrasian) equilibrium.” (a ﬁxed point).
If we move from microeconomics to macroeconomics, we must pay special attention
to time. Now, each individual’s budget set does depend upon time. At each point in
time, he can change his asset portfolio so that he smoothes out his consumption plan
and/or production plan over time. If you know exactly when you die, there is no problem.
Because you just leave no money when you die, unless you want to leave some money
to your kids (i.e., altruistic preferences). This is called the ﬁnite time horizon problem.
What if you might live longer than you expected with no money left? Then, what do
you do? So, in reality, you don’t know exactly when you die. This situation can be
formulated as the inﬁnite time horizon problem. Do you see why? To deal with the
inﬁnite horizon problem, we use the transversality condition as the terminal condition of
the feasible set. Moreover, his optimization must be taken into account time. You can
also analogously deﬁne a sequence of competitive equilibria of the economy.
How can we summarize what we discussed above? Given a (per capita) consump-
tion streamc , (per capita) capital accumulationk , (per capita) GDP stream
f(k ) , capital depreciation rate δ, the population growth rate n, (per capita) con-
sumption growth g, instantaneous utility function of the representative consumer u(·),
the eﬀective discount rate of the representative consumerβ 0, (per capita) wage proﬁle
w , and capital interest rate proﬁler :
1. Find a c such that k = f(k )− c − (δ + g + n)k holds at each t≥ 1 and
t t t t t
k 0 is exogenously given. This is the feasibility question. Any suchc is called
a feasible consumption stream.
2. Find a feasible consumption stream c that maximizes V = e u(c )dt.
t 0 t
This is the problem of optimality. I assume that V ∞.
3. Find a r ,w such that V (the planner’s optimization) is sustained through
t t 0
market economies where k =(r − n− g)k + w − c holds at each t ≥ 1 and
t t t t t
another condition lim λ e k = 0 holds. This latter condition is sometimes
t→∞ t t
called the transversality condition. This is the ﬁxed point problem. This, in fact,
can be done by choosing r = f (k )− δ and w = f(k )− f (k )k at each t≥ 1.
t t t t t t t
7With appropriate re-interpretations, the above is exactly what we had in the begin-
ning except the transversality condition, which is a genuine feature of macroeconomics.
Theorems provide a compact and precise format for presenting the assumptions and
important conclusions of sometimes lengthy arguments, and so help identify immediately
the scope and limitations of the result presented. Theorems must be proved and a proof
consists of establishing the validity of the statement in the theorem in a way that is
consistent with the rules of logic.
2.1.1 Necessity and Suﬃciency
Consider any two statements, p and q. When we say “p is necessary for q,” we mean
that p must be true for q to be true. For q to be true requires p to be true, so whenever
q is true, we know that p must also be true. So we might have said, instead, that “p is
true if q is true,” or simply that “p is implied by q”(p⇐ q).
Suppose we know that “p⇐ q” is a true statement. What if p is not true? Because
p is necessary for q, when p is not true, then q cannot be true, either. But doesn’t this
just say that “q not true” is necessary for “p not true”? Or that “not-q” is implied by
“not-p”(¬q⇐¬p). This latter form of the original statement is called the contrapositive
Let’s consider a simple illustration of these ideas. Let p be the statement, “x is an
integer less than 10.” Let q be the statement that “x is an integer less than 8.” Clearly,
p is necessary for q (q⇒ p). If we form the contrapositive of these two statements, the
statement ¬p becomes “x is not an integer less than 10,” and “x is not an integer less
than 8.” Then, observe that¬q⇐¬p. However,¬p⇐¬q is false. The value of x could
well be 9.
The notion of necessity is distinct from that of suﬃciency. When we say “p is
suﬃcient for q,” we mean that whenever p holds, q must hold. We can say, “p is true
only if q is true,” or that “p implies q”(p⇒ q). Once again, whenever the statement
9“p⇒ q” is true, the contrapositive statement, “¬q⇒¬p” is also true.
Two implications, “p ⇐ q” and “p ⇒ q,” can both be true. When this is so, I say
that “p is necessary and suﬃcient for q,” or “p is true if and only if q is true,” or “p iﬀ
q.” When “p is necessary and suﬃcient for q,” we say that the statements p and q are
equivalent and write “p⇔ q.”
To illustrate brieﬂy, suppose that p and q are the following statements:
• p≡ “X is yellow,”
• q≡ “X is a lemon.”
Certainly, if X is a lemon, then X is yellow. Here, p is necessary for q. At the same
time, just because X is yellow does not mean that it must be a lemon. It could be a
banana. So p is not suﬃcient for q.
2.1.2 Theorems and Proofs
Mathematical theorems usually have the form of an implication or an equivalence, where
one or more statements are alleged to be related in particular ways. Suppose we have
the theorem “p ⇒ q.” Here, p is the assumption and q is the conclusion. To prove a
theorem is to establish the validity of its conclusion given the truth of its assumption,
and several methods can be used to do that.
1. In a constructive proof, we assume that p is true, deduce various consequences of
that, and use them to show that q must also hold. This is also sometimes called a
direct proof, for obvious reasons.
2. In a contrapositive proof, we assume that q does not hold, then show that p cannot
hold. This approach takes advantage of the logical equivalence between the claims
“p⇒ q” and “¬q⇒¬p” noted earlier, and essentially involves a constructive proof
of the contrapositive to the original statement.
3. In a proof by contradiction, the strategy is to assume that p is true, assume that q
is not true, and attempt to derive a logical contradiction. This approach relies on
the fact that p⇒ q or¬q always is true and if “p⇒¬q” is false, then p⇒ q must
4. In a proof by mathematical induction, I have a statement H(k) which does depend
upon a natural number k. What I want to show is that a statement H(k) is true
for each k=1,2,... . First, I show that H(1) is true. This step is usually easy to
establish. Next, I show that H(k)=⇒ H(k +1), i.e., if H(k) is true, then H(k+1)
is also true. These two steps allows me to claim that I am done.
If I assert that p is necessary and suﬃcient for q, or that “p⇔ q,” we must give a
proof in “both directions.” That is, both “p ⇒ q” and “q ⇒ p” must be established
before a complete proof of the assertion has been achieved.
10It is important to keep in mind the old saying that goes, “Proof by example is no
proof.” Suppose the following two statements are given:
• p≡ “x is a student,”
• q≡ “x has red hair.”
Assume further that we make the assertion “p⇒ q.” Then clearly ﬁnding one student
with red hair and pointing him out to you is not going to convince you of anything.
Examples are good for illustrating but typically not for proving.
Finally, a sort of converse to the old saying about examples and proofs should be
noted. Whereas citing a hundred examples can never prove that a certain property
always holds, citing one solitary counterexample can disprove that the property always
holds. For instance, to disprove the assertion about the color of students’ hair, you need
simply point out one student with brown hair. A counterexample proves that the claim
cannot always be true because you have found at least one case where it is not.
2.2 Set Theory
A set is any collection of elements. Sets of objects will usually be denoted by capital
letters, A,S,T for example, while their members by lower case, a,s, t for example (English
or Greek). A set S is a subset of another set T if every element of S is also an element
of T. We write S⊂ T.If S⊂ T, then x∈ S⇒ x∈ T. The set S is a proper subset of T
if S⊂ T and S =
T; sometimes one writes S T in this case. Two sets are equal sets if
they each contain exactly the same elements. We write S = T whenever x∈ S⇒ x∈ T
and x ∈ T ⇒ x ∈ S. The number of elements in a set S, its cardinality, is denoted
S. The upside down “A,”∀, means “for all,” while the backward “E,”∃ means “there
A set S is empty or is an empty set if it contains no elements at all. It is a subset
of “every” set. For example, if A = x x =0,x 1, then A is empty. We denote
the empty set by the symbol ∅. The complement of a set S in a universal set U is the
set of all elements in U that are not in S and is denoted S . For any two sets S and T
in a universal set U, we deﬁne the set diﬀerence denoted S\T, as all elements in the set
S that are not elements of T. Thus, we can think S = U\S. The symmetric diﬀerence
ST =(S\T)∪ (T\S) is the set of all elements that belong to exactly one of the sets S
and T. Note that if S = T, then ST =∅.
For two sets S and T, we deﬁne the union of S and T as the set S∪T≡x x∈Sor x∈ T.
We deﬁne the intersection of S and T as the set S∩ T≡x x ∈ S and x∈ T. Let
Λ≡1,2,3,... be an index set. In stead of writing S ,S ,S ,..., we can write
1 2 3
S . We would denote the union of all sets in the collection by S , and the
λ λ∈Λ λ
intersection of all sets in the collection as S .
11The following are some important identities involving the operations deﬁned above.
• A∪ B = B∪ A, (A∪ B)∪ C = A∪ (B∪ C),A∪∅ = A
• A∩ B = B∩ A, (A∩ B)∩ C = A∩ (B∩ C),A∩∅ =∅
• A∪(B∩C)=(A∪B)∩(A∪C),A∩(B∪C =(A∩B)∪(A∩C) (Distribute laws)
• A\(B∪ C)=(A\B)∩ (A\C),A\(B∩ C)=(A\B)∪ (A\C) (De Morgan’s laws)
• AB = BA, (AB)C = A(BC),A∅ = A
The collection of all subsets of a set A is also a set, called the power set of A and
denoted byP(A). Thus, B∈P(A)⇐⇒ B⊂ A.
Example 2.1 Let A =a,b,c. Then,P(A)=∅,a,b,c,a,b,a, c,b, c,a,b,c.
The previous argument reveals its stance that the order of the elements in a set
speciﬁcation does not matter. In particular,a,b =b,a. However, on many occasions,
one is interested in distinguishing between the ﬁrst and the second elements of a pair.
One such example is the coordinates of a point in the x− y-plane. These coordinates
are given as an ordered pair (a,b) of real numbers. The important property of ordered
pairs is that (a,b)=(c,d) if and only if a = c and b = d. The product of two sets S and
T is the set of “ordered pairs” in the form (s,t), where the ﬁrst element in the pair is a
member of S and the second is a member of T. The product of S and T is denoted
S× T≡(s,t) s∈ S, t∈ T.
The set of real numbers is denoted by the special symbol R and is deﬁned as
Any n-tuple, or vector, is just an n dimensional ordered tuple (x ,... ,x ) and can
be thought of as a “point” in n dimensional Euclidean space. This space is deﬁned as
the set product
R ≡ R×···× R≡(x ,... ,x ) x ∈ R,i=1,... ,n.
1 n i
Often, we want to restrict our attention to a subset of R , called the “nonnegative
orthant” and denoted R , where
R ≡(x ,... ,x ) x ≥ 0,i=1,... ,n⊂ R .
1 n i
Furthermore, we sometimes talk about the strictly “positive orthant” of R
R ≡(x ,... ,x ) x 0,i=1,... ,n⊂ R .
1 n i
Any ordered pair (s,t) associates an element s∈ S to an element t∈ T.Any collection
of ordered pairs is said to constitute a binary relation between the sets S and T. Many
familiar binary relations are contained in the product of one set with itself. For example,
let X be the closed unit interval, X =0,1. Then the binary relation≥ consists of all
ordered pairs of numbers in X where the ﬁrst one in the pair is greater than or equal to
the second one. When, as here, a binary relation is a subset of the product of one set
X with itself, we say that it is a relation on the set X. A binary relation R on X is
represented by the subset of X× X, i.e.,R⊂ X× X. We can build more structure for
a binary relation on some set by requiring that it possesses certain properties.
Deﬁnition 2.1 A relation R in X is reﬂexive if xRx for all x∈ X.
For example,≥ and = on R are reﬂexive, while is not.
Deﬁnition 2.2 A relationR on X is complete if, for all elements x and y in X, xRy
For example, ≥ on R is complete, while and = are not. Note that R on X is
reﬂexive if it is complete.
Deﬁnition 2.3 A relation R on X is transitive if, for any three elements x,y, and
z∈ X, xRy and yRz implies xRz.
For instance, all≥,=, on R are transitive.
Deﬁnition 2.4 A relation R on X is symmetric if xRy implies yRx and it is anti-
symmetric if xRy and yRx implies x = y.
For example, = is symmetric, while≥ and are not. However,≥ is anti-symmetric,
while is not as well. A relationR is said to be a partial ordering on X if it is reﬂexive,
transitive, and anti-symmetric. If a partial ordering is complete, it is called a linear
ordering. For instance, the relation≥ in R is a linear ordering.
For n ≥ 2, the less-than-or-equal-to relation ≥ on R is deﬁned by (x ,... ,x ) ≥
(y ,... ,y ) if and only if x ≥ y for k =1,... ,n. There is also a strict inequality
1 n k k
relation , which is given by (x ,... ,x ) (y ,... ,y ) if and only if x y for all
1 n 1 n k k
2.3.1 Preference Relations
I now talk a little bit about economics. Here I apply the concept of relations to the
consumer choice problem. The number of commodities is ﬁnite and equal to n. Each
commodity is measured in some inﬁnitely divisible units. Let x =(x ,... ,x ) ∈ R
be a consumption bundle. Let R be a consumption set that is the set of bundles the
consumer can conceive. We represent the consumer’s preferences by a binary relation,
, deﬁned on the consumption set, R .If x x , we say that “x is at least as good as
x ,” for this consumer.
Deﬁnition 2.5 The binary relation on X is said to be strict preference relation
if, x x if and only if x x but x x.
Deﬁnition 2.6 The binary relation ∼ on X is said to be indiﬀerence relation if,
x∼ x if and only if x x and x x.
Exercise 2.1 Show the following:
1. on R is reﬂexive if it is complete.
2. on R is not symmetric.
3. ∼ on R is symmetric.
A function is a relation that associates each element of one set with a single, unique
element of another set. We say that the function f is a mapping, map,or transformation
from one set D to another set R and write f : D → R. We call the set D the domain
and the set R the range of the mapping. If y is the point in the range mapped into by
the point x in the domain, we write y = f(x). In set-theoretic terms, f is a relation from
D to R with the property that for each x∈ D, there is exactly one y∈ R such that xfy
(x is related to y via f).
The image of f is that set of points in the range into which some point in the domain
is mapped, i.e.,
I≡y y = f(x) for some x∈ D⊂ R.
The inverse image of a set of points S⊂ I is deﬁned as
f (S)≡x x∈ D, f(x)∈ S .
The graph of the function f is the set of ordered pairs
G≡(x,y) x∈ D, y = f(x).
If f(x)= y, one also writes x→ y. The squaring function s : R→ R, for example,
can then be written as s : x→ x . Thus, → indicates the eﬀect of the function on an
element of the domain. If f : A→ B is a function and S⊂ A, the restriction of f to S
is the function f deﬁned by f (x)= f(x) for every x ∈ S. There is nothing in the
deﬁnition of a function that prohibits more than one element in the domain from being
mapped into the same element in the range. If, however, every point in the range is
assigned to “at most” a single point in the domain, the function is said to be one-to-one,
that is, for all x,x ∈ D, whenever f(x)= f(x ), then x = x . If the image is equal to
the range - if for every y ∈ R, there is x ∈ D such that f(x)= y, the function is said
to be onto. If a function is one-to-one and onto (sometimes called bijective), then an
inverse function f : R→ D exists that is also one-to-one and onto. The composition
of a function f : A→ B and a function g : B → C is the function g◦ f : A→ C given
by (g◦ f)(a)= g(f(a)) for all a∈ A.
Exercise 2.2 Show that f(x)= x is not a one-to-one mapping.
2.4.1 Least Upper Bound Principle
A set S of real numbers is bounded above if there exists a real number b such that b≥ x
for all x∈ S. This number b is called an upper bound for S. A set that is bounded above
has many upper bounds. A least upper bound for the set S is a number b that is an
upper bound for S and is such that b ≤ b for every upper bound b. The existence of a
least upper bound is a basic and non-trivial property of the real number system.
Fact 2.1 (Least Upper Bound Principle) Any nonempty set of real numbers that is
bounded above has a least upper bound.
This principle is rather an axiom of real numbers. A set S can have at most one least
∗ ∗ ∗ ∗
upper bound, because if b and b are both least upper bounds for S, then b ≤ b and
1 2 1 2
∗ ∗ ∗ ∗ ∗
b ≤ b , which thus implies that b = b . The least upper bound b of S is often called
2 1 1 2
the supremum of S. We write b = supS and b = sup x.
Example 2.2 The set S =(0,5), consisting of all x such that 0 x 5, has many
upper bounds, some of which are 100, 6.73, and 5. Clearly no number smaller than 5
can be an upper bound, so 5 is the least upper bound. Thus, supS =5.
A set S is bounded below if there exists a real number a such that x≥ a for all x∈ S.
The number a is a lower bound for S. A set S that is bounded below has a greatest
∗ ∗ ∗
lower bound a , with the property a ≤ x for all x∈ S, and a ≥ a for all lower bounds
∗ ∗ ∗
a. The number a is called the inﬁmum of S and we write a = inf S or a = inf x.
Thus, we summarize
• supS = the least number greater than or equal to all numbers in S; and
• inf S = the greatest number less than or equal to all numbers in S.
Theorem 2.1 Let S be a set of real numbers and b a real number. Then supS = b if
and only if the following two conditions are satisﬁed:
1. x≤ b for all x∈ S.
2. For each ε 0, there exists an x∈ S such that xb − ε.
Proof of Theorem 2.1:(=⇒) Since b is an upper bound for S, by deﬁnition,
property 1 holds, that is, x≤ b for all x∈ S. Suppose, on the other hand, that there
∗ ∗∗ ∗
is some ε 0 such that x ≤ b − ε for all x ∈ S. Deﬁne b = b − ε. This implies
∗∗ ∗∗ ∗
that b is also an upper bound for S and b b . This contradicts our hypothesis that
b is a least upper bound for S.(⇐=) Property 1 says that b is an upper bound for
S. Suppose, on the contrary, that b is not a least upper bound. That is, there is some
other b such that x ≤ b b for all x ∈ S. Deﬁne ε = b − b. Then, we obtain that
x≤ b − ε for all x∈ S. This contradicts property 2.
Topology in R
3.1 Sequences on R
A sequence is a function k → x(k) whose domain is the set 1,2,3,... of all pos-
itive integers. I denote the set of natural numbers by N = 1,2,.... The terms
x(1),x(2),... ,x(k),... of the sequence are usually denoted by using subscripts: x ,x ,... ,x ,... .
1 2 k
We shall use the notationx , or simplyx , to indicate an arbitrary sequence of
real numbers. A sequencex of real numbers is said to be
1. nondecreasing if x ≤ x for k=1,2,...
2. strictly increasing if x x for k=1,2,...
3. nonincreasing if x ≥ x for k=1,2,...
4. strictly decreasing if x x for k =1,2,...
A sequence that is nondecreasing or nonincreasing is called monotone. A sequence
x is said to converge to a number x if x becomes arbitrarily close to x for all
suﬃcient large k. We write lim x = x or x → x as k→∞. The precise deﬁnition
k→∞ k k
of convergence is as follows:
Deﬁnition 3.1 The sequence x converge to x if for every ε 0, there exists a
natural number N such that x − x ε for all k N . The number x is called
ε k ε
the limit of the sequence x .A convergent sequence is one that converges to some
Note that the limit of a convergent sequence is unique. A sequence that does not
converge to any real number is said to diverge. In some cases we use the notation
lim x even if the sequencex is divergent. For example, we say that x →∞ as
k→∞ k k k
k→∞. A sequencex is bounded if there exists a number M such thatx≤ M for
all k =1,2,... . It is easy to see that every convergent sequence is bounded:If x → x,
by the deﬁnition of convergence, only ﬁnitely many terms of the sequence can lie outside
the interval I =(x− 1,x + 1). The set I is bounded and the ﬁnite set of points from the
17sequence that are not in I is bounded, sox must be bounded. On the other hand, is
every bounded sequence convergent? No. For example, the sequencey =(−1) is
bounded but not convergent.
Theorem 3.1 Every bounded monotone sequence is convergent.
Proof of Theorem 3.1: Suppose, without loss of generality, that x is nonde-
creasing and bounded. Let b be the least upper bound of the set X =x k=1,2,...,
and letε 0 be an arbitrary number. Theorem 2.1 already showed that there must be a
term x of the sequence for which x b − ε. Because the sequence is nondecreasing,
b −εx ≤ x for all kN. But the x are all less than or equal to b because of
N k k
boundedness, so we have b −ε x ≤ b . Thus, for any ε 0, there exists a number
N such thatx − b ε for all kN. Hence,x converges to b .
Theorem 3.2 Suppose that the sequences x and y converge to x and y, respec-
1. lim (x ± y )= x± y
k→∞ k k
2. lim (x · y )= x· y
k→∞ k k
3. lim (x /y )= x/y, assuming that y
=0 for all k and y
k→∞ k k k
Exercise 3.1 Prove Theorem 3.2.
Letx be a sequence. Consider a strictly increasing sequence of natural numbers
k k k ···
1 2 3
and form a new sequence y , where y = x for j =1,2,... . The sequence
j j k
y =x is called a subsequence ofx .
j j k j k
Theorem 3.3 Every subsequence of a convergent sequence is itself convergent, and has
the same limit as the original sequence.
Proof of Theorem 3.3: It is trivial.
Theorem 3.4 If the sequence x is bounded, then it contains a convergent subse-
Proof of Theorem 3.4: Since x is bounded, we can assume that there exists
some M ∈ R such that x≤ M for all k ∈ N. Let y = supx k ≥ n for n ∈ N.
By construction, y is a nonincreasing sequence because the set x k ≥ n shrinks
as n increases. The sequence y is also bounded because −M ≤ y ≤ M. Theorem
3.1 already showed that the sequence y is convergent. Let x = lim y . By the
n n→∞ n
18deﬁnition of y , we can choose a term x n from the original sequencex (with k ≥ n)
n k k n
satisfyingy − x 1/n.
x− x =x− y + y − x ≤x− y +y − x x− y +1/n.
k n n k n n k n
n n n
This shows that x → x as n→∞.
3.1.2 Cauchy Sequences
I have deﬁned the concept of convergence of sequences. Then, a natural question arises as
to how we can check if a given sequence is convergent. The concept of Cauchy sequence,
indeed, enables us to do so.
Deﬁnition 3.2 A sequence x of real numbers is called a Cauchy sequence if for
every ε 0, there exists a natural number N such thatx − x ε for all m,n N .
ε m n ε
The theorem below is a characterization of convergent sequences.
Theorem 3.5 A sequence is convergent if and only if it is a Cauchy sequence.
Proof of Theorem 3.5:(=⇒) Suppose that x converges to x. Given ε 0,
we can choose a natural number N such that x − x ε/2 for all nN. Then, for
x − x =x − x + x− x ≤x − x +x− x ε/2+ ε/2= ε.
m n m n m m
Therefore, x is a Cauchy sequence. (⇐=) Suppose that x is a Cauchy sequence.
First, we shall show that the sequence is bounded. By the Cauchy property, there is a
number M such thatx −x 1 forkM. Moreover, the ﬁnite setx ,x ,... ,x
k M 1 2 M−1
is clearly bounded. Hence, x is bounded. Theorem 3.4 showed that the bounded
sequencex has a convergent subsequencex . Let x = lim x . Becausex is a
k k j k k
Cauchy sequence, for everyε 0, there is a natural number N such thatx −x ε/2
for all m,n N. If we take J is suﬃciently large, we havex − xε/2 for alljJ.
Then for kN and j maxN,J,
x − x =x − x + x − x≤x − x +x − xε/2+ ε/2= ε
k k k k k k k
j j j j
Hence x → x as k→∞.
Exercise 3.2 Consider the sequence x with the generic term
1 1 1 1
x = + +··· + =
2 2 2 2
1 2 k i
19Prove that this sequence is a Cauchy sequence. Hint:
1 1 1
+ +··· +
2 2 2
(n+1) (n+2) (n + k)
1 1 1
+ +··· +
n(n+1) (n + 1)(n+2) (n + k− 1)(n + k)
1 1 1 1 1 1
= − + − +··· + −
n n+1 n+1 n+2 n + k− 1 n + k
Exercise 3.3 Prove that a sequence can have at most one limit. Use proof by contra-
diction. Namely, you ﬁrst suppose, by way of contradiction, that there are two limit
3.1.3 Upper and Lower Limits
Let x be a sequence that is bounded above, and deﬁne y = supx k ≥ n for
k n k
n =1,2,... . Each y is a ﬁnite number and y is a nonincreasing sequence. Then
either lim y exists or is−∞. We call this limit the upper limit (or lim sup) of the
sequencex , and we introduce the following notation:
lim supx = lim (supx k≥ n)
If x is not bounded above, we write lim sup x = ∞. Similarly, if x is
k k k
bounded below, its lower limit (or lim inf, is deﬁned as
lim inf x = lim (infx k≥ n)
Ifx is not bounded below, we write lim inf x =−∞.
k k→∞ k
Theorem 3.6 If the sequence x is convergent, then
lim supx = lim inf x = lim x .
k k k
k→∞ k→∞ k→∞
On the other hand, if lim supx = lim inf x , then x is convergent.
k→∞ k k→∞ k k
I omit the proof of Theorem 3.6.
Exercise 3.4 Determine the limsup and lim inf of the following sequences.
1. x =(−1)
2. x = (−1) (2 + 1/k)+1