The study of sequences of real numbers is a fascinating and important part of mathematics. It is a central topic in analysis, a branch of mathematics that deals with continuous functions and their properties. A sequence is simply a function whose domain is the set of positive integers, and whose range is a set of real numbers. Sequences of real numbers have many interesting and useful properties, and they are used to model a wide range of mathematical and real-world phenomena. In this book, we will delve into the properties of sequences of real numbers and explore their connections with other areas of mathematics. We will start with a gentle introduction to sequences, including the definitions and notation used in the study of sequences. We will then move on to more advanced topics, such as convergence and divergence of sequences, Cauchy sequences, and subsequences. One of the central ideas in the study of sequences is the concept of convergence. A sequence is said to converge if its terms become arbitrarily close to a fixed real number as the index of the sequence increases. We will explore the different types of convergence and their properties, as well as the notion of limit, which is a fundamental concept in analysis. In addition to convergence, we will also study the properties of divergent sequences, which are sequences that do not converge. We will examine the relationship between convergence and divergence and their connection with real-valued functions. Throughout the book, we will use a friendly and engaging tone, making the material accessible to a wide range of readers, including students, mathematicians, and anyone with an interest in mathematics. Whether you are a beginner or an expert, you will find something of interest in this book. Chapter 1: Introduction to Sequences In this chapter, we will introduce the basic concepts and notation used in the study of sequences of real numbers. A sequence is simply a function whose domain is the set of positive integers, and whose range is a set of real numbers. We will start by defining sequences and their terms, and we will explore some basic properties of sequences, such as their limits and bounds. We will also introduce the notation used to represent sequences, including the use of the Greek letter n to represent the index of a sequence. This is a crucial piece of notation, as it allows us to express the properties of a sequence in a concise and easily understood way. Finally, we will look at some examples of sequences, including arithmetic and geometric sequences, and we will explore some of their properties. This will provide a foundation for the more advanced topics we will study later in the book. Definition of Sequence A sequence of real numbers is a function whose domain is the set of positive integers, and whose range is a set of real numbers. The elements of the sequence are referred to as terms, and they are denoted by a variable (usually "a_n") with a subscrip

1. "The Fascinating World of Real Number Sequences"
Book Outline:
Chapter 1: Introduction to Sequences
Chapter 2: Convergence and Divergence of Sequences
Chapter 3: Cauchy Sequences
Chapter 4: Subsequences and Limit Superior/Inferior
Chapter 5: Monotone and Bounded Sequences
Chapter 6: The Bolzano-Weierstrass Theorem
Chapter 7: The Heine-Borel Theorem
Chapter 8: Nested Intervals and Cantor's Theorem
Chapter 9: Continuity and Uniform Continuity of Real-Valued Functions
Chapter 10: The Extreme Value Theorem
Chapter 11: The Intermediate Value Theorem
Chapter 12: Connections with Series
Chapter 13: Applications in Calculus
Chapter 14: Topological Properties of Real Numbers
Chapter 15: Further Topics and Open Problems
Introduction :
The study of sequences of real numbers is a fascinating and important part of
mathematics. It is a central topic in analysis, a branch of mathematics that deals with
continuous functions and their properties. A sequence is simply a function whose

2. domain is the set of positive integers, and whose range is a set of real numbers.
Sequences of real numbers have many interesting and useful properties, and they are
used to model a wide range of mathematical and real-world phenomena.
In this book, we will delve into the properties of sequences of real numbers and explore
their connections with other areas of mathematics. We will start with a gentle
introduction to sequences, including the definitions and notation used in the study of
sequences. We will then move on to more advanced topics, such as convergence and
divergence of sequences, Cauchy sequences, and subsequences.
One of the central ideas in the study of sequences is the concept of convergence. A
sequence is said to converge if its terms become arbitrarily close to a fixed real number
as the index of the sequence increases. We will explore the different types of
convergence and their properties, as well as the notion of limit, which is a fundamental
concept in analysis.
In addition to convergence, we will also study the properties of divergent sequences,
which are sequences that do not converge. We will examine the relationship between
convergence and divergence and their connection with real-valued functions.
Throughout the book, we will use a friendly and engaging tone, making the material
accessible to a wide range of readers, including students, mathematicians, and anyone
with an interest in mathematics. Whether you are a beginner or an expert, you will find
something of interest in this book.
Chapter 1: Introduction to Sequences
In this chapter, we will introduce the basic concepts and notation used in the study of
sequences of real numbers. A sequence is simply a function whose domain is the set of
positive integers, and whose range is a set of real numbers. We will start by defining
sequences and their terms, and we will explore some basic properties of sequences,
such as their limits and bounds.

3. We will also introduce the notation used to represent sequences, including the use of
the Greek letter n to represent the index of a sequence. This is a crucial piece of
notation, as it allows us to express the properties of a sequence in a concise and easily
understood way.
Finally, we will look at some examples of sequences, including arithmetic and geometric
sequences, and we will explore some of their properties. This will provide a foundation
for the more advanced topics we will study later in the book.
Definition of Sequence
A sequence of real numbers is a function whose domain is the set of positive integers,
and whose range is a set of real numbers. The elements of the sequence are referred to
as terms, and they are denoted by a variable (usually "a_n") with a subscript
representing the position of the term in the sequence. For example, the sequence "a_1,
a_2, a_3, ... " represents the first, second, third, etc. terms of the sequence. The goal of
the study of sequences is to understand the properties of the sequence as a whole,
including its convergence and divergence properties, limit behaviors, and connections
with other areas of mathematics.
Examples of Sequence
An example of a sequence is the sequence of square numbers, where each term is the
square of the corresponding positive integer:
a_1 = 1^2 = 1
a_2 = 2^2 = 4
a_3 = 3^2 = 9
a_4 = 4^2 = 16
...

4. This sequence can be represented by the formula a_n = n^2, where n is the positive
integer representing the position of the term in the sequence. Another example of a
sequence is the Fibonacci sequence, where each term is the sum of the two preceding
terms:
a_1 = 1
a_2 = 1
a_3 = a_1 + a_2 = 2
a_4 = a_2 + a_3 = 3
a_5 = a_3 + a_4 = 5
...
This sequence can be represented by the formula a_n = a_{n-1} + a_{n-2}.
Real Life Examples Of a Sequence
I. Stock prices: The daily closing prices of a stock can be considered as a
sequence, where each term represents the stock price on a particular day.
II. Weather data: The temperature readings taken at regular intervals (e.g. every
hour) can be considered as a sequence, where each term represents the temperature
at that particular time.
III. Sports statistics: The scores of a team in a particular sport over a period of time
can be considered as a sequence, where each term represents the score in a specific
game.

5. IV. Population growth: The population of a city or country over time can be
considered as a sequence, where each term represents the population at a particular
point in time.
V. Interest rates: The interest rates on a loan over a period of time can be
considered as a sequence, where each term represents the interest rate in a specific
time period.
VI. DNA sequence: The sequence of nucleotides (A, C, G, T) in a DNA molecule can
be considered as a sequence, where each term represents a specific nucleotide.
These are just a few examples, but sequences can be found in many areas of science,
engineering, and everyday life.
Here are some examples of sequences of real numbers that might be
asked in a CSIR-NET Mathematical Science exam:
i. Arithmetic sequences: These are sequences in which each term is obtained by
adding a constant to the previous term. For example, the sequence 3, 6, 9, 12, ... is an
arithmetic sequence with common difference 3.
ii. Geometric sequences: These are sequences in which each term is obtained by
multiplying the previous term by a constant. For example, the sequence 2, 4, 8, 16, ... is
a geometric sequence with common ratio 2.

6. iii. Power sequences: These are sequences in which each term is obtained by
raising a fixed number to a positive integer power. For example, the sequence 1, 2, 4, 8,
... is a power sequence with base 2.
iv. Harmonic sequences: These are sequences in which each term is obtained by
taking the reciprocal of the corresponding term in another sequence. For example, the
sequence 1, 1/2, 1/3, 1/4, ... is the harmonic sequence of the natural numbers.
v. Exponential sequences: These are sequences in which each term is obtained by
raising a fixed number (called the base) to a variable power. For example, the sequence
2^0, 2^1, 2^2, 2^3, ... is an exponential sequence with base 2.
vi. Logarithmic sequences: These are sequences in which each term is obtained by
taking the logarithm of the corresponding term in another sequence. For example, the
sequence log_2(1), log_2(2), log_2(4), log_2(8), ... is the logarithmic sequence of the
power sequence with base 2.
vii. Triangular numbers: These are sequences in which each term is equal to the
sum of the first n positive integers for some positive integer n. For example, the
sequence 1, 3, 6, 10, ... is the sequence of triangular numbers.
viii. Perfect square numbers: These are sequences in which each term is a perfect
square. For example, the sequence 1, 4, 9, 16, ... is the sequence of perfect square
numbers.
ix. Perfect cube numbers: These are sequences in which each term is a perfect
cube. For example, the sequence 1, 8, 27, 64, ... is the sequence of perfect cube
numbers.
x. Fibonacci numbers: These are sequences in which each term is the sum of the
two preceding terms, starting with the initial terms 0 and 1. For example, the sequence
0, 1, 1, 2, 3, 5, 8, ... is the sequence of Fibonacci numbers.

7. xi. Prime numbers: These are sequences in which each term is a prime number. For
example, the sequence 2, 3, 5, 7, 11, 13, 17, ... is the sequence of prime numbers.
xii. Factorial numbers: These are sequences in which each term is equal to the
product of all positive integers up to that number. For example, the sequence 1, 2, 6, 24,
120, ... is the sequence of factorial numbers.
xiii. Bernoulli numbers: These are sequences of rational numbers that appear in the
Taylor series expansions of certain functions. For example, the sequence 1, -1/2, 1/6, 0,
-1/30, 0, ... is the sequence of Bernoulli numbers.
xiv. Euler numbers: These are sequences of real numbers that appear in the Taylor
series expansions of certain functions. For example, the sequence 1, 0, -1, 0, 5, 0, ... is
the sequence of Euler numbers.
xv. Catalan numbers: These are sequences of natural numbers that arise in
combinatorics and number theory. For example, the sequence 1, 1, 2, 5, 14, 42, 132, ...
is the sequence of Catalan numbers.
xvi. Stirling numbers: These are sequences of integers that arise in combinatorics
and the study of permutations and combinations. For example, the sequence 1, 0, 1, 0,
-1, 0, ... is a sequence of Stirling numbers.
xvii. Bell numbers: These are sequences of natural numbers that count the number of
partitions of a set. For example, the sequence 1, 1, 2, 5, 15, 52, 203, ... is the sequence
of Bell numbers.
xviii. Bessel numbers: These are sequences of real numbers that appear in the Taylor
series expansions of certain functions, such as Bessel functions. For example, the
sequence 1, -1/3, 0, 1/45, 0, ... is a sequence of Bessel numbers.

8. Range of Sequence
The range of a sequence of real numbers is the set of all values that can be taken by
the terms of the sequence. In other words, it is the set of all real numbers that are
output by the sequence. The range of a sequence can be finite or infinite, and it can be
either closed or open, depending on the properties of the sequence.
For example, if the sequence is defined by the formula a_n = n^2, the range of the
sequence is the set of all positive squares, {1, 4, 9, ...}. If the sequence is defined by the
formula a_n = 1/n, the range of the sequence is the interval (0, infinity). In both cases,
the range of the sequence is infinite.
It is important to note that the range of a sequence can be different from the set of all
terms of the sequence. For example, a sequence may have terms that are negative, but
its range may still be positive. Similarly, a sequence may have terms that are not real
numbers, but its range may still consist of real numbers. The range of a sequence can
be found by analyzing the properties of the sequence and the values that the terms can
take.
Here are some examples to illustrate the concept of the range of a
sequence:
I. The sequence {1, 2, 3, 4, 5, ...} has a range of [1, infinity).
II. The sequence {1/2, 1/4, 1/8, 1/16, ...} has a range of (0, 1].
III. The sequence {-1, 1, -1, 1, -1, ...} has a range of [-1, 1].
IV. The sequence {2^n | n is a positive integer} has a range of [2, infinity).
V. The sequence {sin(n)} where n is a positive integer has a range of [-1, 1].

9. These are just a few examples to show the concept of the range of a sequence. It is
important to note that the range of a sequence depends on the definition of the
sequence and the values that the terms can take. In some cases, it may be possible to
find the range of a sequence analytically, while in others it may be more difficult and
require more advanced techniques.
Bounded Sequence
A bounded sequence is a sequence of real numbers that is confined to a specific
interval, meaning that all terms of the sequence lie between two fixed real numbers,
known as the lower and upper bounds. In mathematical notation, a sequence {a_n} is
bounded if there exists real numbers M and N such that M <= a_n <= N for all n.
A sequence is said to be lower bounded if there exists a real number M such that M <=
a_n for all n. Similarly, a sequence is said to be upper bounded if there exists a real
number N such that a_n <= N for all n.
For example, the sequence {1/n} is a bounded sequence, with lower bound 0 and upper
bound 1. The sequence {-n} is lower bounded by -infinity, but not upper bounded. The
sequence {n^2} is upper bounded by infinity, but not lower bounded.
It is important to note that a bounded sequence does not necessarily have a finite
range. The range of a sequence is the set of all values that can be taken by the terms of
the sequence, while the bounds of a sequence are the specific real numbers that define
the interval within which the terms of the sequence lie.
Certainly, here are some examples of bounded sequences:
I. The sequence {1, 2, 3, 4, 5, ...} is a bounded sequence with lower bound 1 and
upper bound infinity.
II. The sequence {-2, -1, 0, 1, 2} is a bounded sequence with lower bound -2 and
upper bound 2.
III. The sequence {1/n} where n is a positive integer is a bounded sequence with
lower bound 0 and upper bound 1.

10. IV. The sequence {sin(n)} where n is a positive integer is a bounded sequence with
lower bound -1 and upper bound 1.
V. The sequence {(1 + 1/n)^n} where n is a positive integer is a bounded sequence
with lower bound e and upper bound e.
Here are some remarks on bounded sequences:
A. Boundedness is a necessary condition for many important concepts and results
in mathematics, such as the existence of limits, convergences, and subsequential limits.
B. A bounded sequence is not necessarily convergent, meaning that the terms of
the sequence may not approach a specific limit as the index increases. However, any
convergent sequence is bounded.
C. The least upper bound (lub) of a bounded sequence is also called its supremum,
and the greatest lower bound (glb) of a bounded sequence is also called its infimum.
D. A sequence can be bounded above, bounded below, or both, but it is considered
bounded if and only if it is bounded above and below.
E. The bounds of a sequence are unique if they exist, but it is possible for a
sequence to have no bounds, in which case it is said to be unbounded.
Monotonic Sequence
A monotonic sequence of real numbers is a sequence of real numbers in which either
every number is greater than its predecessor or every number is less than its
predecessor. In other words, a monotonic sequence of real numbers is either increasing
or decreasing. For example, the sequence 1.5, 2.0, 2.5, 3.0, 3.5 is an increasing
monotonic sequence of real numbers, while the sequence 3.5, 3.0, 2.5, 2.0, 1.5 is a
decreasing monotonic sequence of real numbers. The concept of monotonic sequences

11. of real numbers is important in mathematics and has many applications in fields such as
computer science and data science.
Here is an example of a monotonic sequence of real numbers:
Consider the sequence: 2.3, 2.6, 2.9, 3.2, 3.5. This is an increasing monotonic
sequence of real numbers because every term is greater than its predecessor.
Another example of a decreasing monotonic sequence of real numbers is: 7.5, 7.2, 6.9,
6.6, 6.3. This sequence decreases continuously, with every term being less than its
predecessor.
It's worth noting that a monotonic sequence of real numbers can also converge,
meaning that as the terms get larger or smaller, they eventually approach a limiting
value. This property can be useful in optimization problems and stability analysis in
computer science and data science.
Here is a real-life example of a monotonic sequence of real numbers:
Consider the stock price of a company over time. If the stock price of the company
increases continuously over time, it forms a monotonic sequence of real numbers that is
increasing. For example, the stock price of the company could be $10, $11, $12, $13,
$14. Every term in the sequence is greater than its predecessor, making it an increasing
monotonic sequence.
Similarly, if the stock price of the company decreases continuously over time, it forms a
monotonic sequence of real numbers that is decreasing. For example, the stock price of
the company could be $14, $13, $12, $11, $10. Every term in the sequence is less than
its predecessor, making it a decreasing monotonic sequence.

12. In finance, understanding the trend of stock prices over time can be important for
making investment decisions. Monotonic sequences can be useful in analyzing stock
prices to identify patterns and make predictions about future trends.
Subsequence
A subsequence of real numbers is a sequence of real numbers that can be obtained by
selecting some or all of the terms of another sequence of real numbers in a specific
order. In other words, a subsequence is a smaller sequence that is derived from a larger
sequence by removing some of its terms. For example, given the sequence of real
numbers 1.5, 2.0, 2.5, 3.0, 3.5, the subsequences 1.5, 2.5, 3.5 and 2.0, 3.0 are both
valid subsequences of the original sequence.
Subsequences are important in mathematics and have many applications in fields such
as computer science and data science, particularly in areas such as optimization and
pattern recognition. For example, in computer science, algorithms are often designed to
identify the longest increasing or decreasing subsequence in a given sequence of real
numbers, which can be useful in a variety of applications such as finding the most
profitable stock prices in a stock market.
Complementry Subsequence
A complementary subsequence of real numbers is a sequence of real numbers that can
be obtained by removing some of the terms of another sequence of real numbers. In
other words, a complementary subsequence is the portion of a sequence that is left
over after some of its terms have been removed. For example, given the sequence of
real numbers 1.5, 2.0, 2.5, 3.0, 3.5, the complementary subsequence 1.5, 2.5, 3.5 can
be obtained by removing the term 2.0 from the original sequence.
Complementary subsequences are important in mathematics and have many
applications in fields such as computer science and data science. For example, in
computer science, algorithms are often designed to identify the longest increasing or
decreasing complementary subsequence in a given sequence of real numbers, which
can be useful in a variety of applications such as finding the most profitable stock prices
in a stock market.
Here is an example of a complementary subsequence of real
numbers:

13. Consider the sequence of real numbers: 1.5, 2.0, 2.5, 3.0, 3.5.
A valid complementary subsequence of this sequence could be 1.5, 2.5, 3.5, which can
be obtained by removing the term 2.0 from the original sequence.
Another valid complementary subsequence of this sequence could be 2.0, 3.0, which
can be obtained by removing the terms 1.5, 2.5, and 3.5 from the original sequence.
It's worth noting that a complementary subsequence is not a subsequence in the
traditional sense, as it does not involve selecting terms from the original sequence.
Instead, a complementary subsequence involves removing terms from the original
sequence to create a new sequence.
Remarks
Complementary subsequences can be used to analyze and understand the structure of
a given sequence of real numbers. For example, by identifying the longest increasing or
decreasing complementary subsequence, one can gain insights into the overall pattern
and trend of the original sequence.
Complementary subsequences can be used in optimization problems. For example, one
could use complementary subsequences to find the minimum or maximum value of a
given sequence of real numbers.
Complementary subsequences can be used in pattern recognition problems. For
example, one could use complementary subsequences to identify the most common or
rare pattern in a given sequence of real numbers.
Notes:-
some notes on complementary subsequences of real numbers:

14. A. Complementary subsequences are different from traditional subsequences in
that they involve removing terms from a sequence, rather than selecting terms from a
sequence.
B. Complementary subsequences can be used to analyze the structure and pattern
of a given sequence of real numbers.
C. Complementary subsequences can be used in optimization problems, such as
finding the minimum or maximum value of a given sequence.
D. Complementary subsequences can also be used in pattern recognition problems,
such as identifying the most common or rare patterns in a given sequence.
E. Complementary subsequences are a useful tool in fields such as computer
science, data science, and mathematics.
Limit Point of a Sequence
A limit point of a sequence of real numbers is a real number that the sequence can
approach as closely as desired by taking terms from the sequence. In other words, a
limit point is a value that the sequence can get arbitrarily close to, but may never
actually equal.
The concept of limit points is an important one in mathematics and has numerous
applications in fields such as calculus, analysis, and topology. For example, in calculus,
the limit of a function at a point is defined as the limit point of the sequence of values of
the function as the input approaches that point.
To formally define a limit point, we use the idea of neighborhoods. A neighborhood of a
real number x is an open interval that contains x and is arbitrarily small. A limit point of a
sequence (a_n) is a real number L such that for every neighborhood of L, there is a
natural number N such that for all n ≥ N, the terms a_n are in the neighborhood of L.
This means that the terms of the sequence can be made arbitrarily close to L by taking
terms from the sequence beyond a certain point.

15. The limit point of a sequence is often denoted by using the symbol "lim" followed by the
sequence and the limit point, for example: lim_{n→∞} a_n = L. This notation means that
as n approaches infinity, the terms of the sequence a_n approach the limit point L.
Here is an example of a limit point of a sequence of real numbers:
Consider the sequence (a_n) = 1/n, where n is a natural number. The terms of this
sequence get arbitrarily close to 0 as n increases, but they never actually equal 0.
Therefore, 0 is a limit point of the sequence (a_n). We can formally write this as:
lim_{n→∞} 1/n = 0
This means that as n approaches infinity, the terms of the sequence 1/n approach 0 as
closely as desired, making 0 a limit point of the sequence.
Another example is the sequence (a_n) = (-1)^n. This sequence alternates between -1
and 1 and does not have a limit point. However, this sequence does have a limit along
every convergent subsequence.
some remarks on limit points of a sequence of real numbers:
A. Limit points are a key concept in mathematical analysis and play an important
role in understanding the behavior of sequences of real numbers.
B. A sequence may have multiple limit points, but it can also have no limit points.
C. The concept of limit points is closely related to the idea of convergent sequences,
which are sequences that approach a specific limit point as the index increases.
D. The limit point of a sequence can be used to determine whether the sequence is
bounded or unbounded, as well as to analyze the rate of convergence of the sequence.

16. E. Limit points are also useful in the study of topology, where they are used to
define the closure of a set of points.
F. In some cases, the limit point of a sequence can be used to determine the value
of an infinite sum or product.
G. Limit points are important in fields such as calculus, analysis, and topology, and
have numerous applications in areas such as engineering, physics, and computer
science.
some notes on limit points of a sequence of real numbers:
A. Limit points provide information about the behavior of sequences as the index
increases without bound.
B. A sequence may have multiple limit points or none at all, depending on its
behavior.
C. A limit point of a sequence is a real number that the terms of the sequence can
approach as closely as desired, but may never actually equal.
D. The concept of limit points is closely related to the idea of convergent sequences,
which are sequences that approach a specific limit point.
E. Limit points are useful in the study of topology and play an important role in
understanding the closure of a set of points.
F. Limit points are also important in the study of calculus and analysis, as they
provide information about the behavior of sequences and the rate of convergence.
G. The limit point of a sequence can be used to determine whether the sequence is
bounded or unbounded.
H. In some cases, the limit point of a sequence can be used to determine the value
of an infinite sum or product.
Here is a real-life example of a limit point of a sequence of real
numbers:

17. Consider the temperature of a cup of coffee over time. As time goes by, the temperature
of the coffee decreases and approaches a limit point, which is the room temperature.
Let (a_n) be the sequence of temperatures of the coffee at time n, where n is measured
in minutes. Then, as n increases without bound, the terms of the sequence (a_n)
approach the limit point T, where T is the room temperature. We can formally write this
as:
lim_{n→∞} a_n = T
This means that as n approaches infinity, the terms of the sequence (a_n) approach T
as closely as desired, making T a limit point of the sequence.
In this example, the limit point T represents the temperature that the cup of coffee
approaches as it cools down. By analyzing the limit point of the sequence, we can
determine the final temperature of the coffee. This type of analysis is useful in many
real-life applications, such as predicting the temperature of an object over time or
determining the rate of cooling.
Bolzano Weirestrass Theorem :
The Bolzano-Weierstrass Theorem is a fundamental result in real analysis that states:
Every bounded sequence of real numbers has at least one convergent subsequence.
In other words, if (a_n) is a bounded sequence of real numbers, then there exists a
subsequence (a_{n_k}) of (a_n) that converges to a limit. This limit is called the limit
point of the sequence (a_{n_k}).
The Bolzano-Weierstrass Theorem is an important result because it provides a sufficient
condition for a sequence to have a convergent subsequence. This means that even if a
sequence does not converge to a limit, it still may have a subsequence that converges.
The theorem is named after the mathematicians Bernard Bolzano and Karl Weierstrass,
who both made important contributions to the study of real analysis. The theorem is
widely used in various fields of mathematics, including topology, metric spaces, and
complex analysis.

18. In conclusion, the Bolzano-Weierstrass Theorem is a fundamental result in real analysis
that provides a sufficient condition for a sequence to have a convergent subsequence.
This theorem has numerous applications and is an important tool for understanding the
behavior of sequences of real numbers.
Here's a real-life example of the Bolzano-Weierstrass Theorem:
Imagine you're driving on a long straight road and you pass a series of mile markers.
Let (a_n) be the sequence of the distances (in miles) you are from the starting point
after passing the nth mile marker, where n is the number of mile markers you have
passed.
Since the road is straight and you're driving at a constant speed, the sequence (a_n) is
bounded. That is, there exists a positive real number M such that for all n, 0 <= a_n <=
M.
By the Bolzano-Weierstrass Theorem, the sequence (a_n) has a convergent
subsequence. This means that there exists a subsequence (a_{n_k}) of (a_n) such that
as k increases without bound, the terms of the sequence (a_{n_k}) approach a limit.
In this example, the limit of the convergent subsequence (a_{n_k}) represents the final
distance you will be from the starting point after driving the entire distance. By analyzing
the limit of the convergent subsequence, you can determine your final location on the
road.
This example demonstrates how the Bolzano-Weierstrass Theorem can be used in
real-life situations to make predictions about the behavior of sequences of real
numbers.
Here are some notes on the Bolzano-Weierstrass Theorem:
The theorem applies to bounded sequences of real numbers. A sequence is bounded if
there exists a positive real number M such that for all n, 0 <= a_n <= M.

19. The theorem states that every bounded sequence of real numbers has at least one
convergent subsequence.
A subsequence of a sequence is a new sequence formed by selecting certain terms of
the original sequence.
A sequence is said to converge to a limit if, as the terms of the sequence are listed in
order, the terms become arbitrarily close to a certain real number.
The limit of a convergent subsequence is called a limit point of the original sequence.
The Bolzano-Weierstrass Theorem provides a sufficient condition for a sequence to
have a convergent subsequence, but it does not guarantee that the original sequence
converges.
The theorem is named after the mathematicians Bernard Bolzano and Karl Weierstrass,
who both made important contributions to the study of real analysis.
The theorem is widely used in various fields of mathematics, including topology, metric
spaces, and complex analysis.
In conclusion, the Bolzano-Weierstrass Theorem is a fundamental result in real analysis
that states every bounded sequence of real numbers has at least one convergent
subsequence. This theorem has numerous applications and is an important tool for
understanding the behavior of sequences of real numbers.
Convergent Sequence
A convergent sequence of real numbers is a sequence (a_n) such that as n increases
without bound, the terms of the sequence approach a limit, which is a real number.

20. Formally, a sequence (a_n) is said to converge to a limit L if, for every positive real
number ε, there exists a positive integer N such that for all n >= N, |a_n - L| < ε. This
means that, for large enough values of n, the terms of the sequence are arbitrarily close
to L.
A convergent sequence is said to be a Cauchy sequence if, for every positive real
number ε, there exists a positive integer N such that for all m, n >= N, |a_m - a_n| < ε.
This means that the terms of the sequence become arbitrarily close to each other for
large values of m and n.
Convergent sequences and their limits play a fundamental role in real analysis and have
numerous applications in various areas of mathematics. For example, convergent
sequences are used to define the limit of a function and to study the continuity and
differentiability of functions.
Here are some results related to convergent sequences of real
numbers:
I. Every convergent sequence is bounded: This means that there exists a positive
real number M such that for all n, 0 <= a_n <= M.
II. Every Cauchy sequence is convergent: This means that every Cauchy sequence
converges to a limit, which is a real number.
III. The limit of a convergent sequence is unique: This means that if a sequence
converges to two different real numbers, then it cannot be a convergent sequence.

21. IV. The limit of a convergent sequence is well-defined: This means that the limit
does not depend on the choice of subsequence.
V. The limit of a convergent sequence is a limit point of the sequence: This means
that for every positive real number ε, there exists a positive integer N such that for all n
>= N, |a_n - L| < ε, where L is the limit of the sequence.
VI. The limit of a convergent sequence is a cluster point of the sequence: This
means that for every positive real number ε, there exists a positive integer N such that
for all n >= N, the interval (L - ε, L + ε) contains an element of the sequence, where L is
the limit of the sequence.
VII. The limit of a convergent sequence is a fixed point of the function f(x) = x: This
means that the limit of a convergent sequence (a_n) is the unique real number L such
that f(L) = L, where f(x) = x.
These results help to further understand the behavior of convergent sequences and
their limits, and they are used in various areas of mathematics to study the convergence
of sequences, functions, and series.
Theorem (Bolzano-Weierstrass): Every bounded sequence of real
numbers has a convergent subsequence.
Proof: Let (a_n) be a bounded sequence of real numbers, and let M be a positive real
number such that for all n, |a_n| <= M. By the Axiom of Completeness, there exists a
monotone increasing subsequence (a_{n_k}) such that the limit L = lim_k→∞ a_{n_k}
exists. Suppose that L is not equal to a_{n_k} for any positive integer k. Then, there
exists a positive real number ε such that |L - a_{n_k}| >= ε for all positive integers k.

22. Since (a_{n_k}) is a monotone increasing subsequence, it follows that |a_{n_k} - a_{n_j}|
<= M for all positive integers k, j, where k < j. Hence, for all positive integers k, j, the
interval (L - M, L + M) contains an element of the sequence (a_{n_j}), and this
contradicts the definition of the limit L. Therefore, L must equal a_{n_k} for some
positive integer k, and this completes the proof.
This theorem states that every bounded sequence of real numbers has a convergent
subsequence, which implies that every sequence has a convergent subsequence. This
is a powerful result that has numerous applications in various areas of mathematics.
Theorem (Cauchy Convergence Criterion): A sequence of real
numbers (a_n) is convergent if and only if for every positive real
number ε, there exists a positive integer N such that for all positive
integers m, n, if m, n >= N, then |a_m - a_n| < ε.
Proof: Suppose that (a_n) is a convergent sequence, and let ε be a positive real
number. Let L = lim_n→∞ a_n. Then, there exists a positive integer N such that for all
positive integers n, if n >= N, then |a_n - L| < ε/2. For all positive integers m, n, if m, n
>= N, then
|a_m - a_n| <= |a_m - L| + |L - a_n| < ε/2 + ε/2 = ε.
Therefore, (a_n) satisfies the Cauchy Convergence Criterion.
Conversely, suppose that (a_n) satisfies the Cauchy Convergence Criterion. Let ε be a
positive real number. Then, there exists a positive integer N such that for all positive
integers m, n, if m, n >= N, then |a_m - a_n| < ε. Let L be the limit of the sequence
(a_n). Then, for all positive integers n,
|a_n - L| <= |a_n - a_m| + |a_m - L| < ε + |a_m - L|,

23. where m >= N. Hence, for all positive integers n, if n >= N, then |a_n - L| < ε, which
implies that (a_n) is convergent.
This theorem provides a useful criterion for checking the convergence of a sequence of
real numbers. If a sequence satisfies the Cauchy Convergence Criterion, then it is
convergent. Conversely, if a sequence is convergent, then it satisfies the Cauchy
Convergence Criterion.
Divergent Sequence
A divergent sequence of real numbers is a sequence of real numbers where the terms
do not approach a specific limit as the number of terms increases. In other words, the
terms in a divergent sequence do not converge to a specific real number. Divergent
sequences can increase or decrease without bound, or oscillate between two or more
values.
One example of a divergent sequence of real numbers is the sequence (n), where n is a
positive integer. As n approaches infinity, the terms in the sequence become arbitrarily
large and do not approach a specific limit, making it a divergent sequence.
Another example of a divergent sequence of real numbers is the sequence (-1)^n. As n
approaches infinity, the terms in the sequence alternate between -1 and 1 without
converging to a specific limit, making it a divergent sequence.
These examples show that there are many different types of divergent sequences of
real numbers, and that they can behave in many different ways.
Some important remarks on divergent sequences of real numbers
include:
A. Divergent sequences do not have a limit: Unlike convergent sequences, which
have a specific limit, divergent sequences do not have a limit.

24. B. Divergent sequences can grow arbitrarily large: One common characteristic of
divergent sequences is that they can grow arbitrarily large as the number of terms
increases.
C. Divergent sequences can oscillate between two or more values: Another
common characteristic of divergent sequences is that they can oscillate between two or
more values without converging to a specific limit.
D. Every non-convergent sequence is divergent: If a sequence of real numbers is
not convergent, it must be divergent.
E. Divergent sequences are not unique: There are many different types of divergent
sequences, and they can behave in many different ways.
F. These remarks highlight some of the key features of divergent sequences of real
numbers, and help to provide a better understanding of what it means for a sequence to
be divergent.
Some key notes on divergent sequences of real numbers include:
Definition: A divergent sequence of real numbers is a sequence of real numbers where
the terms do not approach a specific limit as the number of terms increases.
Lack of a limit: Unlike convergent sequences, which have a specific limit, divergent
sequences do not have a limit.
Growth without bound: One common characteristic of divergent sequences is that they
can grow arbitrarily large as the number of terms increases.

25. Oscillation: Another common characteristic of divergent sequences is that they can
oscillate between two or more values without converging to a specific limit.
Non-convergent sequences are divergent: Every non-convergent sequence is divergent.
Uniqueness: There are many different types of divergent sequences, and they can
behave in many different ways.
These notes provide a succinct overview of what it means for a sequence of real
numbers to be divergent, and highlight some of the key features of this type of
sequence.