Lecture Notes - Real Analysis II
MAT31M1
Compiled by R. Shukla (Dr)
Faculty of Natural Sciences
Department of Mathematical Sciences & Computing
Walter Sisulu University
2022
,Contents
1 Countable and uncountable sets 1
1.1 Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Countable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Topology of Real Line 8
2.1 Open and Closed Sets in R . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 The Characterization of Closed Sets . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 The Characterization of Open Sets . . . . . . . . . . . . . . . . . . . . . 17
2.4 Subsequences and the Bolzano-Weierstrass Theorem . . . . . . . . . . . . 20
2.4.1 The Existence of Monotone Subsequences . . . . . . . . . . . . . 23
2.4.2 The Bolzano-Weierstrass Theorem . . . . . . . . . . . . . . . . . . 24
2.5 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
i
,Chapter 1
Countable and uncountable sets
1.1 Finite and Infinite Sets
When we count the elements in a set, we say “one, two, three, . . . ,” stopping when we
have exhausted the set. From a mathematical perspective, what we are doing is defining
a bijective mapping between the set and a portion of the set of natural numbers. If the
set is such that the counting does not terminate, such as the set of natural numbers
itself, then we describe the set as being infinite.
The notions of “finite” and “infinite” are extremely primitive, and it is very likely that
the reader has never examined these notions very carefully. In this section we will define
these terms precisely and establish a few basic results and state some other important
results that seem obvious but whose proofs are a bit tricky.
Definition 1.1.1. (a) The empty set ∅ is said to have 0 elements.
(b) If n ∈ N, a set S is said to have n elements if there exists a bijection from the set
Nn := {1, 2, . . . , n} onto S.
(c) A set S is said to be finite if it is either empty or it has n elements for some
n ∈ N.
(d) A set S is said to be infinite if it is not finite.
1
, 1.2 Countable Sets
We now introduce an important type of infinite set.
Definition 1.2.1. (a) A set S is said to be denumerable (or countably infinite) if
there exists a bijection of N onto S.
(b) A set S is said to be countable if it is either finite or denumerable.
(c) A set S is said to be uncountable if it is not countable.
From the properties of bijections, it is clear that S is denumerable if and only if there
exists a bijection of S onto N. Also a set S1 is denumerable if and only if there exists
a bijection from S1 onto a set S2 that is denumerable. Further, a set T1 is countable if
and only if there exists a bijection from T1 onto a set T2 that is countable. Finally, an
infinite countable set is denumerable.
2
MAT31M1
Compiled by R. Shukla (Dr)
Faculty of Natural Sciences
Department of Mathematical Sciences & Computing
Walter Sisulu University
2022
,Contents
1 Countable and uncountable sets 1
1.1 Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Countable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Topology of Real Line 8
2.1 Open and Closed Sets in R . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 The Characterization of Closed Sets . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 The Characterization of Open Sets . . . . . . . . . . . . . . . . . . . . . 17
2.4 Subsequences and the Bolzano-Weierstrass Theorem . . . . . . . . . . . . 20
2.4.1 The Existence of Monotone Subsequences . . . . . . . . . . . . . 23
2.4.2 The Bolzano-Weierstrass Theorem . . . . . . . . . . . . . . . . . . 24
2.5 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
i
,Chapter 1
Countable and uncountable sets
1.1 Finite and Infinite Sets
When we count the elements in a set, we say “one, two, three, . . . ,” stopping when we
have exhausted the set. From a mathematical perspective, what we are doing is defining
a bijective mapping between the set and a portion of the set of natural numbers. If the
set is such that the counting does not terminate, such as the set of natural numbers
itself, then we describe the set as being infinite.
The notions of “finite” and “infinite” are extremely primitive, and it is very likely that
the reader has never examined these notions very carefully. In this section we will define
these terms precisely and establish a few basic results and state some other important
results that seem obvious but whose proofs are a bit tricky.
Definition 1.1.1. (a) The empty set ∅ is said to have 0 elements.
(b) If n ∈ N, a set S is said to have n elements if there exists a bijection from the set
Nn := {1, 2, . . . , n} onto S.
(c) A set S is said to be finite if it is either empty or it has n elements for some
n ∈ N.
(d) A set S is said to be infinite if it is not finite.
1
, 1.2 Countable Sets
We now introduce an important type of infinite set.
Definition 1.2.1. (a) A set S is said to be denumerable (or countably infinite) if
there exists a bijection of N onto S.
(b) A set S is said to be countable if it is either finite or denumerable.
(c) A set S is said to be uncountable if it is not countable.
From the properties of bijections, it is clear that S is denumerable if and only if there
exists a bijection of S onto N. Also a set S1 is denumerable if and only if there exists
a bijection from S1 onto a set S2 that is denumerable. Further, a set T1 is countable if
and only if there exists a bijection from T1 onto a set T2 that is countable. Finally, an
infinite countable set is denumerable.
2
