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Escola de Verão UFSC - Introdução a análise

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Real Analysis
Part I - Real Numbers
Prof. Fabio Silva Botelho
January 6, 2018
1 Initial remarks concerning the original edition
This text develops the basic concepts on real analysis in R and Rn. In the first chapter, we present
a careful construction of the real number system through the definition of cut. In the subsequent
chapters, we present some definitions and results concerning metric spaces. We finish such a study
with a formal proof of the Arzela-Ascoli theorem.
Here we emphasize a standard study on real sequences and series is also presented. In these initial
chapters, the results are in general not new and may be found in many other standard books, such
as Walter Rudin [1] and Elon Lages Lima [2, 3], with exception of the new proof for the intermediate
value theorem and the rigorous study on the exponential functions and logarithms.
On the other hand, in the multi-variable part, we highlight the new proofs of the implicit function
theorem for the vectorial case, which is performed through an application of the Banach fixed point
theorem. This latter theorem is also the basis for a new proof of the inverse function theorem for
Rn.
These new proofs are rigorous, but we believe they are relatively easy to follow.
As an application of the implicit function theorem for the vectorial case, we develop our Lagrange
multiplier result.
Once more the proofs are rigorous but very well developed and easy to follow.
Finally, in the last chapter, we present a study on surfaces in Rn which includes differential forms
defined on such surfaces.
We prefer to use the word surface instead of manifold, since we are referring to a special class of
manifolds properly specified.
In this context, we rigorously establish the definition of volume form and formally prove the
algorithm to obtain the standard formulas.
The Stokes theorem in a more abstract fashion is also developed. In the final section we present
an introduction to Riemannian geometry.
Fabio Botelho
Floriano´polis, May 2017.
1
2 Real Numbers
2.1 Introduction
In this chapter we present the construction of the real numbers. We start with the concept of sets
and relations. In the final sections we introduce the concept of countable set and some concerned
applications.
3 Sets and Relations
By a set we shall understand a collection of objects (also called the set elements), without specify-
ing a more formal definition. We shall describe and/or represent a set either by declaring a propriety
satisfied by its elements or in a straightforward fashion, by specifying its elements.
For example, consider the set A, described by the propriety,
A = {x | x is a month of the year with 31 days}.
Or specifying its elements,
A = {January, March, May, July, August, October, December}.
Given a set A, if x is in A (that is, if x is an element of A), we write x ∈ A. If x is not in A, we
denote x 6∈ A.
Other examples of sets:
A = {x | x is a country which shares a border with the USA},
hence
A = { Canada, Mexico }.
B = {x | 3/x is integer},
that is
B = {3,−3, 1,−1}.
3.1 Empty and Unitary Sets
If a set A has no elements, is said to be empty, and in such a case we denote A = ∅.
For example,
A = {x | x is an American state which shares a border with England},
Hence A = ∅.
On the other hand, a set which has only one element, is said to be an unitary one.
2
For example
A = {x | 2x+ 3 = 7},
that is A = {2} (unitary set).
Definition 3.1. Let A,B be sets. We say that A is contained in B if the following propriety holds:
If x ∈ A then x ∈ B.
In such a case we denote A ⊂ B.
Moreover, if A ⊂ B and there exists x ∈ B such that x 6∈ A the inclusion is said to be proper.
Finally, if there exists x ∈ A such that x 6∈ B we say that A is not contained in B and denote
A * B.
Remark 3.2. Let A,B be two sets. If A ⊂ B and B ⊂ A we say that A equals B and write A = B.
3.2 Properties of Inclusion
The inclusion has the following elementary properties:
Proposition 3.3. Let A,B,C be sets. Thus,
1. ∅ ⊂ A,
2. A ⊂ A
3. If A ⊂ B and B ⊂ C then A ⊂ C.
3.3 Parts of a set
Definition 3.4. Let A be a set. We define the set of parts of A, denoted by P(A) by
P(A) = {B : B ⊂ A}.
Hence the set of parts of A is the set whose elements are the subsets of A.
3.4 Union of sets
At this point we assume the set A ⊂ U , where U is a general class of sets. We call U the universal
set of the class in question.
A B
U
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Figure 1: Two sets A and B in a universal set U .
Definition 3.5 (Union). Let A,B be sets. we define the union of A and B, denoted by A ∪B, by
A ∪ B = {x : x ∈ A or x ∈ B}.
A ∪B
A B
U
Figure 2: Union of two sets A and B in a universal set U .
Proposition 3.6 (Properties of union). Let A,B,C be sets. thus,
1. A ∪A = A,
2. A ∪ ∅ = A,
3. A ∪B = B ∪ A,
4. (A ∪B) ∪ C = A ∪ (B ∪ C).
3.5 Intersection
In this subsection we define the intersection between sets and present some concerned properties.
Definition 3.7. Let A,B be sets. We define the intersection of A and B, denoted by A ∩ B, by
A ∩B = {x : x ∈ A and x ∈ B}.
A ∩B
A B
U
Figure 3: Intersection of two sets A and B in a universal set U .
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Proposition 3.8 (Intersection properties). Let A,B,C be sets. Concerning intersection definition,
the following properties hold:
1. A ∩A = A,
2. A ∩ ∅ = ∅,
3. A ∩B = B ∩ A,
4. (A ∩B) ∩ C = A ∩ (B ∩ C).
Definition 3.9 (Disjoint sets). Let A,B be sets. If A ∩B = ∅ then A and B are said to be disjoint
sets.
Proposition 3.10 (Properties involving unions and intersections). Let A,B,C be sets. The following
proprieties are valid concerning unions and intersections:
1. A ∪ (A ∩B) = A,
2. A ∩ (A ∪B) = A,
3. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C),
4. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C),
Proof. we prove just property 3. Observe that x ∈ A∪(B∩C) implies that x ∈ A or (x ∈ B and x ∈
C)
if x ∈ A then x ∈ (A ∪ B) and x ∈ (A ∪ C) so that x ∈ (A ∪B) ∩ (A ∪ C).
if x ∈ B and x ∈ C then x ∈ (A ∪ B) and x ∈ (A ∪ C), so that x ∈ (A ∪ B) ∩ (A ∪ C).
Hence, in any case
A ∪ (B ∩ C) ⊂ (A ∪B) ∩ (A ∪ C).
Conversely, assume x ∈ (A ∪ B) ∩ (A ∪ C). There are two cases to consider. x ∈ A or x 6∈ A If
x ∈ A, then x ∈ A ∪ (B ∩ C).
If x 6∈ A, as x ∈ (A ∪ B) we must have x ∈ B. Similarly as x ∈ A ∪ C, we must have x ∈ C so
that x ∈ B ∩ C ⊂ A ∪ (B ∩ C).
Thus (A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C).
We may conclude that (A ∪ B) ∩ (A ∪ C) = A ∪ (B ∩ C) which completes the proof.
We finish this section presenting the definition of difference between sets.
Definition 3.11 (Difference). Let A,B be sets. We define A minus B, denoted by A \B, by
A \B = {x : x ∈ A and x 6∈ B}.
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A \B
A B
U
Figure 4: Difference A minus B between the sets A and B.
B \ A
A B
U
Figure 5: Difference B minus A between the sets A and B.
Hence we denote Ac = U \ A, that is,
Ac = {x ∈ U : x 6∈ A}.
We refer to Ac as the complement of A relating U .
4 Cartesian product and relations
Definition 4.1 (Cartesian product, relations). Let A, B be sets. We define the cartesian product of
A by B, denoted by A× B by:
A×B = {(x, y) | x ∈ A, y ∈ B}.
Any subset R ⊂ A×B is said to be a relation from A to B.
Definition 4.2 (Function). Let A and B be two sets. A relation f from A to B is said to be a
function if for each x ∈ A there is one and only one y ∈ B such that (x, y) ∈ f. In such a case we
denote f : A→ B and y = f(x).
Definition 4.3. Let f : A → B be a function. We say that f is injective if the following propriety
is valid:
If x1, x2 ∈ A and x1 6= x2 then f(x1) 6= f(x2),
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or equivalently
If x1, x2 ∈ A and f(x1) = f(x2) then x1 = x2.
Definition 4.4. A function f : A→ B is said to be surjective if R(f) = B, where R(f) denotes the
range or image of f , defined by
R(f) = {f(x) : x ∈ A}.
Definition 4.5. Let E ⊂ B. We define the inverse image of E, by f , denoted