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
