REAL ANALYSIS

REAL NUMBERS

INTRODUCTION

Real analysis is, roughly speaking, the modern setting for calculus, ‘real’ alluding to the field of real numbers.

This ext on real numbers discusses of sequence that culminates in the concept of convergence, the fundamental concept of analysis. We shall look at Weierstrass-Bolzano theorem, normally a theorem about bonded sequence, is in essence a property of closed intervals and Cauchy’s Criterion is a test for convergence, especially useful in t he theory of infinite series and finally we shall look at a dissection of convergence into two more general limiting operations.

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REAL NUMBERS     …        …        …        …        …        …        …        …        …        …        2
BOUNDED SEQUENCES   …        …        …        …        …        …        …        …        …        4
ULTIMATELY, FREQUENTLY     …        …        …        …        …        …        …        …        7
NULL SEQUENCE   …        …        …        …        …        …        …        …        …        …        9
CONVERGENT SEQUENCE          …        …        …        …        …        …        …        …        12
SUBSEQUENCES, WEIERSTRASS-BOLZANO THEOREM     …        …        …        …        15
CAUCHY’S CRITERION FOR CONVERGENCE           …        …        …        …        …        …        17
MONOTONE SEQUENCES            …        …        …        …        …        …        …        …        …        18

BOUNDED SEQUENCES

Bounded Sets

V. S. Shipachev 1998 P. 28

A set x is said to be bounded from above (from below) if there is a number c such that the inequality x ≤ c (x ≥ c) is satisfied for any x Є X. in this case the number C is the upper (lower) bound of the set x.

A set which is bounded both from above and from below is said to be bounded. Thus, for instance, any finite interval  is bounded. The interval  is a set bounded from below but not bounded above, and the entire number line  is a set not bounded wither from above or from below.

Any set x bounded from above (from below) evidently has infinitely many upper (lower) bounds. Indeed, if the number c is an upper (lower) bound of the set x, then any number c; which is larger (smaller) than the number c, is also an upper (lower) bound of the set x since the validity of the inequality  implies the validity of the inequality

Definition of bounded sequence
Sterling, K. Berberian 1994 P. 33-34

A sequence (Xn) of real numbers is said to be bounded if the set is bounded.
A sequence that is not bounded is said to be unbounded.

Remark
A sequence (Xn) in  is bounded if and only if there exist a positive number K such that for all n.

Proof

therefore by theorem (1)

Example
Every constant sequence for all n.) is bounded. The sequence is bounded.

Theorem 2
If and are bounded sequences in , the sequences and are also bounded.

Proof
If and then.
and .

Monotonic Sequence theorem
James Stewart 4th  edition P. 734
Every bounded, monotonic sequence is convergent.

Proof
Suppose is an increasing sequence. Since  is bounded, the set = has an upper bound. By the completeness Axion if has a least upper bound L. Given is not an upper bound for S. therefore for some integer N.

So since thus , whenever
So lim .

ULTIMATELY, FREQUENTLY.

Definition
Sterling K. Berberian 1994 P. 35-36

Let (Xn) be a sequence in a set x and let A be a subset of x

1. we say that ultimately if Xn belongs to A from some index onward that is, there is an index N such that for all  symbolically,

1. we say that frequently is for every index N there is an index for which ; symbolically.

Example
Let let  and let then  ultimately.

Proof
Choose an index N such that   then.

Example
For each positive integer n, let Sn be a statement (which may be either true or false).

Let A = {n Є |P : Sn is true}

Theorem 3
With notations as above, one and only one of the following conditions holds:
(1) ultimately
(2)  frequently

Proof
To say that (1) is false means that, whatever index N is proposed, the implication
           
is false, so there must exist an index  for which is precisely the meaning of (2).

For example is (Xn) is an sequence in then either Xn < 5 ultimately, or  frequently, but not both.

NULL SEQUENCE

Definition

Sterling K. Berberian 1994 P. 36-38

A sequence (Xn) in  is said to be null if, for every positive real number  ultimately.
For example, the sequence  is null.

Theorem 4
Let (Xn) and (Yn) be null sequences and let
Then;

1. (Xn) is bounded
2. (CSn) is null
3. (Xn + Yn) is null
4. if (bn) is a bounded sequence then (bnXn) is null
5. if (Zn) is a sequence such that  ultimately, then (Zn) is also null.

Proof

1. with  = 1, we have  ultimately; let N be an index such that  for all n > N. If K is the larger of the numbers.

(1),  _ _ _ _ ,  then  for every positive integer n, thus (Xn) is bounded.

(2). let . Since (Xn) is null, there is an index N1 such that

so if N is the larger of  N1 and N2­ then

(4). let k be a positive real number such that  for all n. Given any  choose an index N such that;  thus  is null.

(2) is a special case of (4)

(5) By assumption, there is an index N1 such that

Given any  choose an index N2 such that
If N = max  then
 thus  is null.

Theorem 5
If  or  then the sequence (an - a) is null.

Poof
If  then

In particular,  and if . Given any  choose an index N such that;
 is null. The case that  is deduced by applying the foregoing to .

Example
Fix  and let Xn = X for all n. The Constant sequence (Xn) is null if and only if x = 0.

Proof

The condition “ ultimately” means  if this happens for every  then x = 0 (for, if ) then  is an embarrassment)

CONVERGENT SEQUENCE

The Concept of a Convergent Sequence

Definition
V.S. Shipachev 1998 P. 36-37
The number a is the limit of the sequence  if for any positive number  there is a number N such that the inequality
             
holds true for n > N.

A sequence which has a limit is said to be convergent. If the sequence  converges and has a number a as its limit, then the symbolic notation is lim  or  as  .

A sequence which is not convergent is divergent,

Example
Using the definition of the limit of a sequence, we shall prove that,

we take an arbitrary number . Since  it follows that to find the values of n which satisfy the inequality  it is sufficient to solve the inequality , whence we find that  

Definition
James Stewart 4th edition P. 729
A sequence  has the limit L and we write    or  as ,

If for every  there is a corresponding integer N such that  where  
If lim   exists, we say the sequence converges (of is convergent). Otherwise, we say the sequence diverges (or s divergent).

Limit laws for sequence
James Stewart 4th edition P. 730
 If  and  are convergent sequences and c is a constant, then
  
  

   if

The squeeze theorem for sequence is if  for  and  then.  

Example
Evaluate    if it exists.

Solution
  

Therefore by the theorem if    then
 .

Theorem 6
The sequence  is convergent if  and divergent for all other values of r   

Definition
A sequence  is called increasing if  for all , that is,  If is called decreasing if  for all . It is called monotonic if it is either increasing or decreasing.

SUBSEQUENCES, WEIERSTRASS-BOLZANO THEOREM

Steling K. Berberian, 1994 P. 43-44

Given a sequence, there are various ways of forming ‘subsequences’ for example, take every other term,  or take all terms for which the index is a prime number,
The general idea is that one is free to discard any terms, as long as infinitely many terms remain.

Remark
Here’s is useful perspective on subsequences.
A sequence  is a set x can be thought of as a function.

 where  =
A sequence of  is obtained by specifying of strictly increasing function.

 and taking the composition function fo 

Thus, writing  we have

Theorem 7 (weierstrass- Bolzano theorem)
Every bounded sequence in  has a convergent subsequence.
Proof
Let  be a bounded sequence of real numbers. By the preceding theorem,  has a monotone subsequence . Suppose, for example, that  is increasing; it is also bounded =, so  for a suitable real number a, and .

Corollary
In a closed interval  every sequence has a subsequence that converges to a point of the interval.

Proof
Suppose . By the theorem, some subsequence is convergent to a point of , say  since  for all k, it follows that  thus

CAUCHY’S CRITERION FOR CONVERGENCE

Theorem 8 (Cauchy’s Criterion)
Steling K. Berberian 1994 P. 48-49

For a sequence  in  the following conditions are equivalent.
(a)  is convergent;
(b) for every  there is an index N such that  whenever m and n are

Proof
(a)  (b): say  if  then  ultimately, say for  if both m and n are  then, by the triangle inequality,

(b)
 (a): Assuming (b), let’s show first that the sequence  is bounded. Choose an index M such that  for all m,  then, for all

therefore the sequence  is bounded, explicitly, if  max  then  for all n.

MONOTONE SEQUENCES

Monotone, monotonic function
Yule Bricks 2006 P. 116.
A function is monotone if it only increases or only decreases, f increases monotonically (is monotonic increasing) if , implies  that =

A function f decreases monotonically (is monotonic decreasing) if
A function f is strictly monotonically increasing is , implies that  >  and strictly monotonically decreasing if , implies that  < .

Definition
James Stewart 4th edition P. 733.

A sequence  is called increasing if  for all  that is,  It is called decreasing if  for all  It is called monotonic if it is either increasing or decreasing.
Example
The sequence  is decreasing because  for all

(The right side is smaller because it has a larger denominator).

Definition
V.S. Shipachev 188 P. 42

The sequence  is said to be increasing if  for all n, non decreasing if  for all n, decreasing if  for all n.

All sequence of these kinds are united by a common title of monotone sequences.
Increasing and decreasing sequences are also called strictly monotone sequences.
Here are some examples of monotone sequences.

1. The sequence  is decreasing and bounded.
2. The sequence  is non increasing and bounded.
3. The sequence 1,2,3……,n,…….. is increasing and bounded.
4. The sequence 1,1,2,2,3,3,…..,n,n,….  Is non decreasing and unbounded.
5. The sequence  is increasing and bounded.

Or ‘a sequence in x’, whose nth term is xn
Various notations are used to indicated sequences, for example

Definition
A sequence  in  is said to be increasing if  that is, if  for all n; increasing if
 and strictly decreasing if  for all n.

A sequence that is either increasing or decreasing is said to be monotone; more precisely, one speaks of sequences that are ‘monotone increasing ‘or ‘ monotone decreasing’.

The genda of this text can be recommended as a foundation of calculus including the fundamental theorem and along was to develop those skills and attitudes that enable us to continue learning mathematics on out own.

Some part of real analysis is an attempt to bring some significant pat of the theory of lebesgue integral a powerful generalization calculus.
Real analysis can also be recommended for the application of applied Abstract algebra and introduction to optimal control theory.

This text gives the fundamentals of real functions and I tried to present the material as completely, strictly and simple as possible. The aim was no to convey certain quantity of knowledge in mathematics but also to arouse an interest in any reader of this text.

Real functions can be used by mathematicians for theoretical material, and therefore for practical purposes

Also it can be applied for mathematical prediction possible for complicated phenomena and mechanisms.

Real functions can be applied to solve specific theoretical and practical problems to stimulate and elaborate of newer abstract methods and branches of mathematics.

James Stewart (1999) CALCULUS, BOSTON.

K. A. Stroud (2006) ENGINEERING MATHEMATICS, NEW YORK

Sterling K. Berberian (1994) A FIRST COURSE IN REAL ANALYSIS, NEW YORK

V. S. Shipachev (1988)HIGHER MATHEMATICS, RUSSIA

Yule Bricks (2006) DICTIONARY OF MATHEMATICS, NEW DELHI