Sent with FAQ on 24.1.2003

Sent With FAQ on 24.1.2003

11.1.2013

Advanced Calculus Unit XXI

sequences of real numbers – Part V

Objectives

From this session a learner is expected to achieve the following

Study the concept of Cauchy sequence
Study Cauchy Convergence Criterion
Familiarize with sequences that are Cauchy
Familiarize with sequences that are not Cauchy
Study that contractive sequences are Cauchy and hence convergent
Learn the definition of properly divergent sequences
Study two comparison theorems that help to verify the divergence of certain sequences.

Contents

1. Introduction

2. Cauchy sequence

3. Cauchy Convergence Criterion

4. Sequences that are not Cauchy

5. Contractive Sequences

6. Properly Divergent Sequences

7. Comparison Theorems

1. Introduction

A Cauchy sequence, named after the French mathematician Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. In this session we also discuss the Cauchy Criterion. We will see that the condition that a sequence of real numbers is Cauchy implies the convergence of the sequence. The advantage of this condition is that it does not require the knowledge of the value of the limit. In this session we will discuss contractive sequences also. A discussion on properly divergent sequences will be made. Some comparison theorems will be discussed.

2. Cauchy Sequence

Definition A sequence of real numbers is said to be a Cauchy sequence if for every there is a natural number such that for all natural numbers , the terms satisfy .

We now show that a convergent sequence is a Cauchy sequence.

Theorem 1 If is a convergent sequence of real numbers, then is a Cauchy sequence.

Proof. If , then, given there is a natural number such that if then . Thus, if and if , then

, using the Triangle Inequality

Since is arbitrary, it follows that is a Cauchy sequence.

We now show that a Cauchy sequence is bounded.

Theorem 2 A Cauchy sequence of real numbers is bounded.

Proof. Let be a Cauchy sequence and let . If and , then . Hence, by the Triangle Inequality we have

that is for . If we set

then it follows that for all .

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3. Cauchy Convergence Criterion

We now present the important Cauchy Convergence Criterion.

Theorem 3 (Cauchy Convergence Criterion) A sequence of real numbers is convergent if and only if it is a Cauchy sequence.

Proof. We have seen, in Theorem 1, that a convergent sequence is a Cauchy sequence.

Conversely, let be a Cauchy sequence; we shall show that is convergent to some real number. ByTheorem 2, the Cauchy sequence is bounded. Therefore, by the Bolzano-Weierstrass Theorem, there is a subsequence of that converges to some real number We claim that converges to

Since is Cauchy sequence, given there is a natural number such that if then

…(1)

Since the subsequence converges to , there is a natural number belonging to the set such that

Since , it follows from Eq.(1) with that

for.

Therefore, if , we have

Since is arbitrary, we infer that . Therefore the sequence is convergent. This completes the proof.

We shall now give some examples of applications of the Cauchy Criterion.

Example1 Show that the sequence is Cauchy and hence is convergent.

Solution

Take

To show that is a Cauchy sequence we note that if is given, then there is a natural number such that . Hence, if then we have and . Therefore it follows that if then

.…(2)

Since is arbitrary, inequality in (2) implies that is a Cauchy sequence; therefore, it follows from the Cauchy Convergence Criterion that it is a convergent sequence.

Example 2 Verify the convergence of the sequence defined by

, andand.

Also find the limit of the sequence.

Solution

By mathematical induction, it can be proved that

for.

Thus, if , using the Triangle Inequality, we obtain

Therefore, given, if is chosen so large that and if , then it follows that . Therefore, is a Cauchy sequence in . Hence by the Cauchy Convergence Criterion the sequence converges to a number .

Since converges to , every subsequence of it also converges to . In particular, the subsequence with odd indices also converges to . By applying induction, it can be seen that

Hence it follows that

and hence

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4. Sequences that are not Cauchy

If a sequence is not Cauchy, then it can be verified using the negation of the definition of Cauchy sequence: That is, a sequence is not Cauchy if there exists such that for every natural number H there exist at least one and at least one such that .

Example 3 The sequence is not a Cauchy sequence.

Let . We have to find an such that for every natural numberH there exist at least one and at least one such that .

For the terms , we observe that if is even, then and If we take , then for any we can choose an even number and let to get

We conclude that is not a Cauchy sequence.

Remark We note that to prove a sequence is a Cauchy sequence, we may not assume a relationship between and , since the required inequality must hold for all . But to prove a sequence is not a Cauchy sequence, we may specify a relation between and as long as arbitratily large values of and can be chosen so that .

Example 4 Show that the sequence diverges, where

Solution

If , then

Since each of these terms exceeds, then

In particular, if we have. This shows that the sequence is not a Cauchy sequence; therefore, by Cauchy Convergence Criterion, is not a convergent sequence.

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5. Contractive Sequences

Definition We say that a sequence of real numbers is contractive if there exists a constant such that and

for all . The number is called the constant of the contractive sequence.

Theorem 4 Every contractive sequence is a Cauchy sequence, and therefore is convergent.

Proof.

Let be a contractive sequence of real numbers. Then there exists a constant such that and

for all .…(3)

Taking in place of , the above gives

Taking in place of , (3) gives

Hence (3) gives

.…(4)

For we have

, by applying the Triangle Inequality

, using (4)

, using the formula for the sum of a geometric progression

.…(5)

Since , we have and so (5) implies that is a Cauchy sequence. Hence by the Cauchy Convergence Criterion, is a convergent sequence. This completes the proof.

6. Properly Divergent Sequences and Oscillatory Sequences

We now discuss sequences that are not convergent.

Definition Let be a sequence of real numbers.

(i) We say that the sequencediverges to , and write , if for every there exists a natural number such that if , then .

(ii) We say that diverges to , and write , if for every there exists a natural number such that if , then .

We say that is properly divergent if or .

RemarkThe symbols and in the above expressions do not represent real numbers.

Definition A sequence of real numbers which is neither convergent nor properly divergent is called an oscillatory sequence.

Example 5Show the sequence is properly divergent.

Solution

If is given, let be any natural number such that . Then for . Hence. Hence the sequence is properly divergent.

Example 6The sequence is an oscillatory sequence, since the sequence is not convergent and since neither diverges to nor diverges to .

Example 7Show the sequence diverges to .

Solution

If is given, letbe any natural number such that . Then for any , we have . Hence and the sequence is diverges to .

Example 8 Show the sequence is properly divergent, where .

Solution

Let , where . If is given, let be a natural number such that . If , we have

, by Bernoulli’s Inequality

Therefore and the sequence is properly divergent.

Monotone Convergence Theorem states that a monotone sequence is convergent if and only if it is bounded. The next result is a reformulation of Monotone Convergence Theorem.

Theorem 5 A monotone sequence of real numbers is properly divergent if and only if it is unbounded. Also,

a) If is an unounded increasing sequence, then .

b) If is an unbounded decreasing sequence, then .

Proof. (a) Suppose that is an increasing sequence. We know that if is bounded, then it is convergent. If is unbounded, then for any there exists such that . But since is increasing, we have for all . Since is arbitrary, it follows that diverges to . i.e., .

Similarly, Part (b) can be proved.

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7. Comparison Theorems

We now discuss two comparison theorems that are useful to show that a sequence is properly divergent.

Theorem 6 (Comparison Theorem) Let and be two sequences of real numbers and suppose that

for all .…(6)

a) If , then

b) If , then .

Proof. (a) If and if is given, then there exists a natural number such that if , then . Hence by the inequality (6), it follows that for all . Since is arbitrary, it follows that .

The statement (b) can be proved similarly.

Remarks

In using Theorem 6 to show that a sequence tends to we need to show that the terms of the sequence are ultimately greater than or equal to the corresponding terms of a sequence that tends to . Similarly, to show that a sequence tends to we need to show that the terms of the sequence are ultimately less than or equal to the corresponding terms of a sequence that tends to .
If condition (6) of the Theorem 6 holds and if , it does not follow that . Similarly, if condition (6) holds and if , it does not follow that .
Theorem 6 remains true if condition (6) is ultimately true; that is, if there exists such that for all .

The following limit comparison theorem is frequently used.

Theorem 7 (Limit Comparison Theorem) Let and be two sequences of positive real numbers and suppose there exists a real number such that

Then if and only if

Proof. Suppose

Then corresponding to there exists a natural number such that

for all

i.e., for all

Since each is positive, the above implies

for all .…(7)

If and if is given, then there exists a natural number such that if , then . Hence by the second inequality in (7), it follows that for all . i.e., for all . Since is arbitrary, it follows that .

If and if is given, then there exists a natural number such that if , then . Hence by the first inequality in (7), it follows that for all . Since is arbitrary, it follows that . This completes the proof.

Remark The conclusion in Theorem 7 need not hold if either or . For example, if and is the constant sequence then , but . Also, if take and , then, and but .

Summary

In this session we have introduced the notion of Cauchy sequences and discussed the Cauchy Criterion. In this session we haveseen that contractive sequences are convergent. A discussion on properly divergent sequences and oscillatory sequence have been made.

Assignments

1.Show that the sequence where is a Cauchy sequence.

2.Show that the sequence where is not a Cauchy sequence.

3. Show that if and are Cauchy sequences, then and are Cauchy sequences.

4.If are arbitrary real numbers and for , show that is convergent. What is its limit?

5. Let be properly divergent and let be such that belongs to . Show that converges to 0.

FAQ

1. State Triangle Inequality.

Answer. For any two real numbers a and b,

2. Whether the symbols and represent real numbers?

Answer. No. The symbols and do not represent real numbers.

3. Is there any relation between Cauchy sequences and bounded sequences?

Answer. A Cauchy sequence of real numbers is bounded. But a bounded sequence need not be a Cauchy sequence.

4. What can you say about the convergence of a Cauchy Sequence?

Answer. If a sequence of real numbers is Cauchy, then it is convergent. Also, a convergent sequence of real numbers is a Cauchy sequence.

5. What can you say about the convergence of an unbounded increasing sequence.

Answer. An unbounded increasing sequence is properly convergent.

QUIZ

1.Pick the False statement.

(a) There are bounded sequences that are not Cauchy.

(b) is not a Cauchy sequence

(d) If is a Cauchy sequence such that is an integer for all , then is a constant sequence.

Ans. (d)

2.Pick the False statement.

(a) A bounded, monotone increasing sequence is a Cauchy sequence.

(b) If and for all , then is a Cauchy sequence.

(c)If are arbitrary real numbers and for , then is not bounded.

(d) If are arbitrary real numbers and for , then is convergent.

Ans. (c)

3.Pick the true statement.

(a)If is an unbounded sequence, then there exists a properly divergent subsequence.

(b) If and for , then is not a contractive sequence.

(d) none of the above.

Ans. (a)

4.Pick the true statement.

(a) If for all then if and only if

(b) If for all then if and only if

(c)If for all then if and only if

(d) If for all then if and only if

Ans. (d)

Glossary

Cauchy Sequence: A sequence of real numbers is said to be a Cauchy sequence if for every there is a natural number such that for all natural numbers , the terms satisfy .

Contractive Sequence: A sequence of real numbers is contractive if there exists a constant such that and

for all . The number is called the constant of the contractive sequence.

Properly Divergent Sequence: A sequence of real numbers

(i) diverges to , denoted by, if for every there exists a natural number such that if , then .

(ii) diverges to , denoted by, if for every there exists a natural number such that if , then .

A sequence of real numbers is properly divergent if or .

Oscilatory Sequence: A sequence of real numbers which is neither convergent nor properly divergent is called an oscillatory sequence.

References:

1. T. M. Apostol, Mathematical Analysis, Narosa Publishing House, New Delhi, 1985.

2. R. R. Goldberg, Real Analysis, Oxford & I.B.H. Publishing Co., New Delhi, 1970.

3. D. Soma Sundaram and B. Choudhary, A First Course in Mathematical Analysis, Narosa Publishing House, New Delhi, 1997.

4. Robert G. Bartle, Donald R. Sherbert, Introduction to Real Analysis, Wiley India Pvt. Ltd., New Delhi.