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The Cauchy CriterionMATH 464/506, Real Analysis

J. Robert Buchanan

Department of Mathematics

Summer 2007

J. Robert Buchanan The Cauchy Criterion

Cauchy Sequences

Definition

A sequence X = (xn) of real numbers is a Cauchy sequence ifit satisfies the Cauchy criterion: for all > 0 there existsH() N such that for all m, n H(), then terms xm, xn satisfy|xm xn| < .

Example(

1n

)

is a Cauchy sequence.

(1 + (1)n) is not a Cauchy sequence.

J. Robert Buchanan The Cauchy Criterion

Cauchy Sequences

Definition

A sequence X = (xn) of real numbers is a Cauchy sequence ifit satisfies the Cauchy criterion: for all > 0 there existsH() N such that for all m, n H(), then terms xm, xn satisfy|xm xn| < .

Example(

1n

)

is a Cauchy sequence.

(1 + (1)n) is not a Cauchy sequence.

J. Robert Buchanan The Cauchy Criterion

Cauchy Sequences

Definition

A sequence X = (xn) of real numbers is a Cauchy sequence ifit satisfies the Cauchy criterion: for all > 0 there existsH() N such that for all m, n H(), then terms xm, xn satisfy|xm xn| < .

Example(

1n

)

is a Cauchy sequence.

(1 + (1)n) is not a Cauchy sequence.

J. Robert Buchanan The Cauchy Criterion

Cauchy vs. Convergent

Lemma

If X = (xn) is a convergent sequence of real numbers, then X isa Cauchy sequence.

Proof.

Lemma

A Cauchy sequence of real numbers is bounded.

Proof.

J. Robert Buchanan The Cauchy Criterion

Cauchy vs. Convergent

Lemma

If X = (xn) is a convergent sequence of real numbers, then X isa Cauchy sequence.

Proof.

Lemma

A Cauchy sequence of real numbers is bounded.

Proof.

J. Robert Buchanan The Cauchy Criterion

Cauchy vs. Convergent

Lemma

If X = (xn) is a convergent sequence of real numbers, then X isa Cauchy sequence.

Proof.

Lemma

A Cauchy sequence of real numbers is bounded.

Proof.

J. Robert Buchanan The Cauchy Criterion

Cauchy vs. Convergent

Lemma

If X = (xn) is a convergent sequence of real numbers, then X isa Cauchy sequence.

Proof.

Lemma

A Cauchy sequence of real numbers is bounded.

Proof.

J. Robert Buchanan The Cauchy Criterion

Cauchy Convergence Criterion

Theorem (Cauchy Convergence Criterion)

A sequence of real numbers is convergent if and only if it is aCauchy sequence.

Proof.

J. Robert Buchanan The Cauchy Criterion

Cauchy Convergence Criterion

Theorem (Cauchy Convergence Criterion)

A sequence of real numbers is convergent if and only if it is aCauchy sequence.

Proof.

J. Robert Buchanan The Cauchy Criterion

Examples

Example

Consider the sequence:

x1 = 1, x2 = 2, and xn =12

(xn2 + xn1) for n > 2.

Consider the sequence

y1 =11!

, y2 =11!

12!

, , yn =11!

12!

+ +(1)n+1

n!,

Consider the sequence

z1 =11

, z2 =11

+12

, , zn =11

+12

+ +1n

,

J. Robert Buchanan The Cauchy Criterion

Examples

Example

Consider the sequence:

x1 = 1, x2 = 2, and xn =12

(xn2 + xn1) for n > 2.

Consider the sequence

y1 =11!

, y2 =11!

12!

, , yn =11!

12!

+ +(1)n+1

n!,

Consider the sequence

z1 =11

, z2 =11

+12

, , zn =11

+12

+ +1n

,

J. Robert Buchanan The Cauchy Criterion

Examples

Example

Consider the sequence:

x1 = 1, x2 = 2, and xn =12

(xn2 + xn1) for n > 2.

Consider the sequence

y1 =11!

, y2 =11!

12!

, , yn =11!

12!

+ +(1)n+1

n!,

Consider the sequence

z1 =11

, z2 =11

+12

, , zn =11

+12

+ +1n

,

J. Robert Buchanan The Cauchy Criterion

Contractive Sequences

Definition

A sequence of real numbers X = (xn) is contractive if thereexists a constant 0 < C < 1 such that

|xn+2 xn+1| C|xn+1 xn|

for all n N.

Theorem

Every contractive sequence is a Cauchy sequence andtherefore is convergent.

Proof.

J. Robert Buchanan The Cauchy Criterion

Contractive Sequences

Definition

A sequence of real numbers X = (xn) is contractive if thereexists a constant 0 < C < 1 such that

|xn+2 xn+1| C|xn+1 xn|

for all n N.

Theorem

Every contractive sequence is a Cauchy sequence andtherefore is convergent.

Proof.

J. Robert Buchanan The Cauchy Criterion

Contractive Sequences

Definition

A sequence of real numbers X = (xn) is contractive if thereexists a constant 0 < C < 1 such that

|xn+2 xn+1| C|xn+1 xn|

for all n N.

Theorem

Every contractive sequence is a Cauchy sequence andtherefore is convergent.

Proof.

J. Robert Buchanan The Cauchy Criterion

Further Results

Corollary

If X = (xn) is a contractive sequence with constant 0 < C < 1and if lim X = L, then

1 |xn L| Cn1

1 C|x2 x1|

2 |xn L| C

1 C|xn xn1|

Proof.

J. Robert Buchanan The Cauchy Criterion

Further Results

Corollary

If X = (xn) is a contractive sequence with constant 0 < C < 1and if lim X = L, then

1 |xn L| Cn1

1 C|x2 x1|

2 |xn L| C

1 C|xn xn1|

Proof.

J. Robert Buchanan The Cauchy Criterion

Application

Example

Approximate the solution 0 < x < 1 to the equation

x3 7x + 2 = 0

J. Robert Buchanan The Cauchy Criterion

Homework

Read Section 3.5.

Page 86: 1, 2, 3, 7, 8 , 9, 11, 14

Boxed problems should be written up separately and submittedfor grading at class time on Friday.

J. Robert Buchanan The Cauchy Criterion