__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.

__CONTENTS__

__PAGE__

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

**CHAPTER ONE**

__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__

If for all n and if (for example) then and , thus by the following.

__Theorem 1__

For real number a,b,c,x:

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.

But the sequence is increasing so for every. Thus, if we have
So since thus , whenever

So lim .

__CHAPTER TWO__

__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

- 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,

- 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}

We say that Sn is true frequently is frequently, and that Sn is true ultimately if ultimately. For example, ultimately (infact, for ) and n is frequently divisible by 5 (infact, for n = 5, n = 10, n = 15, etc).
__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.

__CHAPTER THREE__

__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;

- (Xn) is bounded
- (CSn) is null
- (Xn + Yn) is null
- if (bn) is a bounded sequence then (bnXn) is null
- if (Zn) is a sequence such that ultimately, then (Zn) is also null.

__Proof __

- 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

this proves that is null.

(3). let k be a positive real number such that for all n. Given any choose an index N such that then

thus is null.
(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)

__CHAPTER FOUR__

__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

consequently, we can take the integral part of the number as N, that is

. Then the inequality will be satisfied for all we have thus proved 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.

__CHAPTER FIVE__

__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 .

The weierstrass-Bolzano theorem can be reformulated as a theorem about closed intervals:
__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

__CHAPTER SIX__

__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.

__CHAPTER SEVEN__

__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.

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

__RECOMMENDATON__

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.

__CONCLUSION__

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.

__REFERENCES__

*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__