BSc(H) MATHEMATICS (BOOKLET REAL ANALYSIS)

SECTION-I (REAL NUMBERS & TOPOLOGY OF )

Question 1 State algebraic properties of real numbers.

Question 2

Theorem-: (a) If z and a are elements in with z+a =a, then z=0.

(b) If u and b 0 are elements in with u.b=b,then u=1

(c) If a ,then a.0=0

Question 3

Theorem-: (a) If a 0 and b in such that a.b=1 then b=1/a.

(b) If a.b=0,then either a=o or b=0

Question 4

Theorem-: There does not exist a rational number r such that =2

Definition-: The order properties of

There is a non empty subset P of , called the set of positive real numbers

That satisfies the following properties:

(1) If a, b belong to P,then a+b belongs to P.(2) If a , b belong to P,then ab belong to P.(3) If a belong to ,then exactly one of the following is true:

a P, a=0, -a P

Definition-: let a , b be elements of .

(a) If a-b P ,then we write a>b or b<a

(b) If a-b P ,then we write a b or b a.

PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 5

Theorem-: let a , b , c be any elements of .

(a) If a>b and b>c , then a>c.(b) If a>b,then a+c >b+c. (c) If a >b and c>0, then ca>cb.

If a> b and c<0, then ca<cb.

Question 6

Theorem-: (a) If a and a 0, then >0.

(b)If n , then n>0.

Question 7

Theorem-: If a is such that 0 a< for every >0, then a=0.

Question 8

Theorem-: If ab>0, then either

(1) a>0 and b>0 or(2) a<0 and b<0

Question 9

(a) Let a o and b 0.Then

a < b

(b) If a and b are positive real numbers, then

(c) ,(BERNOULLI’S INEQUALITY).


Question 10

If a , b are real numbers then prove the following.

(a) If a+b=o, then a+b=0 ,then b=-a(b) –(-a)=a(c) (-1)a=-a,(d) (-1)(-1)=1(e) –(a+b)=(-a)+(-b)(f) (–a).(-b)=a.b

(g)

Question 11

If a satisfies a.a=a, prove that either a=o or a=1.

Question 12

,show that.

Question 13

Show that there does not exist a rational number s such that =6

Question 14

Show that there is no rational whose square is 3.

Question 15

Show that sum and product of two rational numbers, is a rational number.

Question 16

(a) Prove that if x is rational and y is irrational then x+y is irrational.(b) If x is non zero rational and y is irrational then x.y is irrational.(c) Show that there is no natural number between 0 and 1


(d) Prove that no natural number can be both even and odd

(e) If c>1 and m , n are natural numbers ,show that .

(f) If 0<c<1 and m , n are natural numbers,show that

Question 17

(a) If a<b and c d, prove that a+c<b+d.(b) If 0<a<b and 0 c d,prove that 0 a.c b.d.

(c) Show that if a>0 , then >0 and .

(d) Show that if a<b , then .

Question 18

(a) If 0<c<1,show that 0<

(b) If 1<c, show that .

(c) If

Definition-: (The set of natural numbers)

There is a set (called set of natural numbers) which has following properties.


Axiom is the basis of mathematical induction.

MATHEMATICAL INDUCTION-:

Let be statements or propositions, then

are true provided

Example 1

Prove that

Example 2

All numbers

Example 3

Show that for all natural numbers .

Example 4

,

Example 5 PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

,

Example 6

Example 7

(a) Prove that .

(b) Prove that ,

(c) Prove that

(d) Prove that

Question

State and prove binomial theorem using mathematical induction.

Definition-: A number is called an algebraic number if it satisfies a polynomial

equation , where

Rational zero theorem.

Suppose that and that is a rational number satisfies the polynomial equation


where .

Write having no common factors,

Then .

Question

Question

Question

can not represent a rational number.

Question

does not represent a rational number.

Question

can not represent a rational number.

Question

(a) Show that are not rational numbers.


(b) Show that are not rational numbers.

(c) Show that are not rational numbers.

Absolute value and real line

Definition-:The absolute value of a real number a, denoted by ,is defined by

Question 19

Theorem- : (a)


Question 20

Theorem-:(Triangle inequality)

If .

Question 21

Corollary-: If

Question 22

Definition-:

Question 23


Theorem-:

.

Question 24

Question 25

Question 26

Qestion 27

.

Question 28

.

Question 29 PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 30

Question 31

Question 32


Question 33









Definition-: (interior point)



SECTION -2(SEQUENCES)

Definition-: A sequence of real numbers (or a sequence in ) is a function

defined on the set of natural numbers whose range is contained in

the set of real numbers.

Notation-:

Definition-: (Fibonacci sequence)

Definition-: A sequence in is said to converse to

Or is said to be a limit of , if for every there exists

such that for all .

Note-: If a sequence has a limit we say that sequence is convergent; if it has no limit we say it is divergent.

Question 64

Theorem-: A sequence in can have at most one limit.


Question 65

Theorem-:

Let be a sequence of real numbers, and let .

The following statements are equivalent

(a) converges to

(b) for every ,there exists

such that ;

(c)for every ,there exists

Such that

(d)for every -neighbourhood of ,there exists

Such that .

Question 66

Show the following

(a)


(b)

(c)

Definition-: if is a sequence

Of real numbers and if m is a given natural number, then the m-tail of is the sequence

Question 67

Theorem-:

m-tail of a sequence is convergent if and only if the sequence is convergent, moreover both shall have the same limit.

Question 68

Theorem-:

Let be a sequence of real numbers and let .

If is a sequence of positive real numbers with

and if for some constant and some we have

,

then it follows that . PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 69

(a)

(b)

(c)

Question 70

For any

Question 71

Use the definition of limit of a sequence to establish the following limits.

(a) (b)


(c) (d)

Question 72

Show that

(a) (b)

(c)

Question 73

Question 74


Question 75

Show that if ,and , then

Question 76

Prove that if , and if , then there exists a natural

number M such that M.

Question 77

Show that

Question 78

Show that for .

Question 80

If show that

such that then


Definition-: A sequence of real numbers is said to be

bounded if such that

Question 81

Theorem-:

Every convergent sequence of real numbers is bounded.

Question 82

Theorem-:

(a) Let and be sequences of real numbers that

conversge to , respectively, and let .Then the

sequences converge to

, respectively.

(b) If converges to and is a sequence of

nonzero real numbers that converges to , then the

sequence converges to .

Question 83

Theorem-:


If is convergent sequence of real numbers and if

for all , then

Question 84

Theorem-:

If and are convergent sequences of real numbers and if

for all then

Question 85

Theorem-:

If is convergent sequence and if for all

then .

Question 86

Theorem-:(squeeze theorem)

Suppose that are such that

then is convergent and .

Question 87 PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Show that

Question 88

Theorem-:

let the sequence converges to . Then the sequence

converges to .

Theorem-:

Let be a sequence of real numbers that converges to

and suppose , then the sequence

.

Theorem-:

Let be a sequence of positive real numbers such that

Question 89

Find limit of the following sequences: PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

(a) (b)

(c) (d)

Question 90

Question 91

Question 92

Determine the following limits.

(a) (b)

Question 93

Question 94


Question 95

Use the squeeze theorem to determine the limits of following.

(a) (b)

Question 96

Question 97

Question 98


(a)give an example of a convergent sequence

(b)give example of a divergent sequence with same property.

Question 99

let be a sequence of positive real numbers such that

show that such that

for all sufficiently large use this to show that .

Question 100

Suppose that is a convergent sequence and is such that for

any there exists M such that M. Does it

follow that ??.

Definition-:

let be a sequence of real numbers.

We say that is increasing if it satisfies the inequalities

We say it is decreasing if it satisfies the inequalities


We say it is monotone if it is either increasing or decreasing.

Question 101

Monotone convergence theorem-:

A monotone sequence of real numbers is convergent if and only if it is bounded. Further :

(a)if is a bounded increasing sequence, then

(b)if is a bounded decreasing sequence, then

Question 102

Show that

Question 103

Let show that is unbounded and hence is divergent.

Question 104

Let be defined inductively by

. Show that .

Question 105


let be the sequence of real numbers defined by

.show that .

Question 106

Let , construct a monotone sequence of real numbers that

converges to .

Question 107

Let . Show that is bounded and

increasing sequence. Also show that .

Question 108

Question 109

Show that is bounded and monotone. Find its limit.

Question 110

Show that is decreasing and bounded below by 2. Find limit. PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

Question 111

Show that is convergent find its limit.

Question 112

show that is convergent. Find its limit.

Question 113

show that is convergent . find its limit.

Question 114

determine if is convergent or divergent.

Question 115

let A be an infinite subset of that is bounded above and let

Show that there exists an increasing sequence

such that .

Question 116

Establish the convergence or the divergence of the sequence


Where

Question 117

Let prove that is increasing and bounded, and hence convergent.

Definition-: let be a sequence of real numbers and

be a strictly increasing sequence of

natural numbers. Then the sequence given by

is called a subsequence of .

Question 118

Theorem-: if a sequence of real numbers converges to a real

number , then any subsequence of PREPARED BY SANJEEV KUMAR SHUKLA BSc(H) MATHS, MSc (maths)( D .U)

also converges to .

Question 119

Theorem-: let be a sequence of real numbers .

Then the following are equivalent.

(a)The sequence does not converge to .

(b)there exists an such that , such that

and

(c)there exists an and a subsequence

Such that .

Divergence criteria-: if a sequence of real numbers has either of the following properties, then is divergent.

(1) has two convergent subsequence and

whose limits are not equal.(2) is unbounded.

Question 120

Monotone subsequence theorem-:


If is a subsequence of real numbers, then there is a subsequence of that is monotone.

Question 121

The Bolzano-Weierstrass theorem

A bounded sequence of real numbers has a convergent subsequence.

Question 122

Theorem-:

Let be a bounded sequence of real numbers and let have the property that every convergent subsequence of converges to .then the sequence converges to .

Question 123

Give an example of an unbounded sequence that has convergent subsequence.

Question 124

Establish the convergence and find the limits of the following sequences:

(a) (b)


(c) (d)

(e)

(f)

Question 125

Prove that limit of a sequence which has the property that its every subsequence has a convergent subsequence that converges to zero,

is zero.

Question 126

let be a bounded sequence and let

and show that there exists a

subsequence of that converges to .

Question 127

Suppose that and that


Show that

Question 128

Show that if is unbounded, then there exists a subsequence

such that .

Question 129

Let be a bounded sequence and let .

Show that if , then there is a subsequence of

That converges to .

Definition-: (Cauchy sequence)

A sequence of real numbers is called Cauchy

If such that

We have .


Lemma -: if is a convergent sequence of real numbers then it is a Cauchy sequence.

Lemma -: Every Cauchy sequence is bounded.

Cauchy convergence criterion-: A sequence of real numbers is convergent if and only if it is Cauchy sequence.

Question 130

Let be defined by

for , discuss the convergence.

Question 131

Let be defined as

Discuss the convergence

Question 132

Give an example of a bounded sequence that is not a Cauchy sequence.

Question 133

Show directly from the definition that the following are Cauchy sequences.


(a) (b)

Question 134

Show by definition that sum and product of two Cauchy sequences is again Cauchy sequence.

Question 135

If show that is a Cauchy sequence

Question 136

for ,show that is convergent. What is its limit??.

Question 137

show that is convergent. What is its limit??.

Definition-:limit (superior and limit inferior)

(a) Let be a sequence of real numbers and

Let then limit superior of the sequence is


defined as .

(b)let be a sequence of real numbers and

Let then limit inferior of the sequence is

Defined as .

Definition.

For a sequence we say that if and only if

such that

Also we say that if and only if

such that

Question

Show that

Theorem-:

Let and be sequences such that


and (finite or ) , Then .

Theorem-:

For a sequence of positive real numbers, we have

if and only if .

Theorem-:

(1) If is an unbounded nondecreasing sequence,

then

(2) If is an unbounded nonincreasing sequence, then

Corollary-:

If is a monotone sequence, then the sequence either converges, diverges to ,or diverges to .

Definition


Let be a sequence of real numbers. A subsequential limit is any real number or symbol or that is the limit of some

subsequence of .

Theorem-:

Let be any sequence of real numbers, and let denote the set of

subsequential limits of .

(1) is nonempty

(2) and .

(3) exists if and only if has exacty one element , namely

.

Theorem-:

Let be the set of all subsequential limits of sequence .

suppose is a sequence in and that , then

belongs to .


Theorem-:

If converges to a positive real number and is any

sequence then .

here we allow the conventions

Theorem-:

Let be any sequence of nonzero real numbers. Then we have

.

Corollary.

If exists, then exists and both limits are same.

Theorem-:

Let be any sequence of real numbers

(1) If is defined(as real numbers ), then

.

(2) If then is defined and

.


SECTION-3(INFINIE SERIES)

Infinite series-: An infinite series is an infinite summation of real

numbers. It is denoted as .

Sequence of partial sum-:

Let be an infinite series then sequence of partial sums of this

infinite series is the sequence defined as

.

Definition-: An infinite series is convergent, divergent according to as its sequence of partial sums Is convergent or divergent.

Question 138

(a) Discuss the convergence of geometric series .

(b) Discuss the convergence of p-series .


Definition-:

we say that a series satisfies Cauchy criterion if its sequence of partial sums is a Cauchy sequence.

Let be an infinite series, and let be sequence of partial sums of this series.

Then in view of above definition we have following.

Theorem-: A series converges if and only if it satisfies Cauchy criterion.

Corollary-: if a series is convergent, then .

Comparison test-:

Let be a series where

(1)If converges and , then converses.

(2) If and , then

Definition-( Absolutely convergent series)

An infinite series is said to be absolutely convergent if and only if

the series is convergent.


Theorem-: Every absolutely convergent series is convergent.

Ratio Test.

A series of nonzero terms

(1) Converges absolutely if ,

(2) Diverges if

(3) Otherwise and the test gives no information.

Root test.

Let be a series and let .

The series

(1) Converges absolutely if

(2) Diverges if

(3) Otherwise and let the test gives no information.

Question 139

Consider the series , discuss the convergence.


Question 140


Question 141

Consider the series , check for the convergence.

Question 142

Consider the series , check for the convergence.

Question 143


Question 144

Determine which of the following series converge. Justify your answer

(a) (b) (c)

(d) (e) (f)


(g) (h) (i)

(j) (k) (l)

(m) (n) (0)

(p) (q) (r)

(s) (t) (u)

The integral test-:

Let be a series of nonnegative terms.

If there is a continuous non-increasing function

Such that

Then series is convergent if

And it diverges if

Alternating series theorem-:


If , then the

alternating series converges.

Question 145

Determine which of the following series converge. justify your answer.

(a) (b) (c)

(d) (e) (f)

(g) (h)


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BSc(H) MATHEMATICS (BOOKLET REAL ANALYSIS)