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 ǫ > 0 there existsH(ǫ) ∈ N such that for all m, n ≥ H(ǫ), then terms xm, xn satisfy|xm − xn| < ǫ .

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 ǫ > 0 there existsH(ǫ) ∈ N such that for all m, n ≥ H(ǫ), then terms xm, xn satisfy|xm − xn| < ǫ .

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 ǫ > 0 there existsH(ǫ) ∈ N such that for all m, n ≥ H(ǫ), then terms xm, xn satisfy|xm − xn| < ǫ .

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

(xn−2 + xn−1) 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

(xn−2 + xn−1) 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

(xn−2 + xn−1) 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| ≤Cn−1

1 − C|x2 − x1|

2 |xn − L| ≤C

1 − C|xn − xn−1|

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| ≤Cn−1

1 − C|x2 − x1|

2 |xn − L| ≤C

1 − C|xn − xn−1|

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