Sequences and Their Algebra, Bolzano Weierstrass Theorem - CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


Sequences and Series of Real
Numbers
We often use sequences and series of numbers without thinking about it. A decimal
representation of a number is an example of a series, the bracketing of a real number
by closer and closer rational numbers gives us an example of a sequence. We want
to study these objects more closely because this conceptual framework will be used
later when we look at functions and sequences and series of functions. First, we will
take on numbers.
Sequences have an ancient history dating back at least as far as Archimedes who
used sequences and series in his \Method of Exhaustion" to compute better values of
¼ and areas of geometric ¯gures.
6.1 The Symbols +1 and ¡1
We often use the symbols +1 and ¡1 in mathematics, including courses in high
school. Weneedtocometosomeagreementaboutthesesymbols. Wewilloftenwrite
1 for +1 when it should not be confusing.
First of all, they arenot real numbers and donot necessarily adhere to the rules
of arithmetic for real numbers. There are times that we \act" as if they do, so we
need to be careful.
We adjoin +1 and ¡1 to R and extend the usual ordering to the set R[
f+1;¡1g. Explicitly, we will agree that ¡1 < a < +1 for every real number
a2R[f+1;¡1g. This gives the extended set with an ordering that satis¯es our
usual properties:
1) If a;b2R[f+1;¡1g, then a·b or b·a.
2) If a·b and b·a, then a=b.
3) If a·b and b·c, then a·c.
We will not extend the usual algebraic structure of the reals to R[f+1;¡1g.
Page 2


Sequences and Series of Real
Numbers
We often use sequences and series of numbers without thinking about it. A decimal
representation of a number is an example of a series, the bracketing of a real number
by closer and closer rational numbers gives us an example of a sequence. We want
to study these objects more closely because this conceptual framework will be used
later when we look at functions and sequences and series of functions. First, we will
take on numbers.
Sequences have an ancient history dating back at least as far as Archimedes who
used sequences and series in his \Method of Exhaustion" to compute better values of
¼ and areas of geometric ¯gures.
6.1 The Symbols +1 and ¡1
We often use the symbols +1 and ¡1 in mathematics, including courses in high
school. Weneedtocometosomeagreementaboutthesesymbols. Wewilloftenwrite
1 for +1 when it should not be confusing.
First of all, they arenot real numbers and donot necessarily adhere to the rules
of arithmetic for real numbers. There are times that we \act" as if they do, so we
need to be careful.
We adjoin +1 and ¡1 to R and extend the usual ordering to the set R[
f+1;¡1g. Explicitly, we will agree that ¡1 < a < +1 for every real number
a2R[f+1;¡1g. This gives the extended set with an ordering that satis¯es our
usual properties:
1) If a;b2R[f+1;¡1g, then a·b or b·a.
2) If a·b and b·a, then a=b.
3) If a·b and b·c, then a·c.
We will not extend the usual algebraic structure of the reals to R[f+1;¡1g.
Instead, when we have to we will discuss the algebra that might involve +1 and/or
¡1. Do not apply any theorem that is stated for the real numbers to the symbols
+1 or¡1.
Thesymbolsmakeitconvenienttoextendournotationaboutintervalstothereal
line.
[a;1)=fx2Rja·xg (a;1)=fx2Rja<xg
(¡1;b]=fx2Rjx·bg (¡1;b)=fx2Rjx<bg:
Occasionally, you will seeR=(¡1;1).
6.2 Sequences
Sequences are, basically, countably many numbers arranged in an ordered set that
may or may not exhibit certain patterns.
De¯nition 6.1 A sequence of real numbers is a function whose domain is a set of
the form fn2Zjn¸mg where m is usually 0 or 1. Thus, a sequence is a function
f:N ! R. Thus a sequence can be denoted by f(m), f(m + 1), f(m + 2), ....
Usually, we will denote such a sequence by fa
i
g
1
i=m
or fa
m
;a
m+1
;a
m+2
;:::g, where
a
i
=f(i). If m =1, we may use the notation fa
n
g
n2N
.
Example 6.1 The sequence f1;1=2;1=3;1=4;1=5;:::g is written as f1=ig
1
i=1
. Keep
in mind that this sequence can be thought of as an ordinary function. In this case
f(n)=1=n.
Example 6.2 Consider the sequence given by a
n
= (¡1)
n
for n¸ 0. This time we
have started the sequence with 1 and the terms look like, f1;¡1;1;¡1;1;¡1;:::g.
Note that this time the function has domainN but the range isf¡1;1g.
Example 6.3 Consider the sequence a
n
= cos
¡
¼n
3
¢
, n 2 N. The ¯rst term in the
sequence is cos
¼
3
=cos60
±
=
1
2
and the sequence looks like
1
2
;¡
1
2
;¡1;¡
1
2
;
1
2
;1;
1
2
;¡
1
2
;¡1;¡
1
2
;
1
2
;1;
1
2
;¡
1
2
;¡1;:::g:
Note that likes its predecessor, the function takes on only a ¯nite number of values,
but the sequence has an in¯nite number of elements.
Example 6.4 If a
n
=n
1=n
, n2N, the sequence is
1;
p
2;3
1=3
;4
1=4
;:::
Page 3


Sequences and Series of Real
Numbers
We often use sequences and series of numbers without thinking about it. A decimal
representation of a number is an example of a series, the bracketing of a real number
by closer and closer rational numbers gives us an example of a sequence. We want
to study these objects more closely because this conceptual framework will be used
later when we look at functions and sequences and series of functions. First, we will
take on numbers.
Sequences have an ancient history dating back at least as far as Archimedes who
used sequences and series in his \Method of Exhaustion" to compute better values of
¼ and areas of geometric ¯gures.
6.1 The Symbols +1 and ¡1
We often use the symbols +1 and ¡1 in mathematics, including courses in high
school. Weneedtocometosomeagreementaboutthesesymbols. Wewilloftenwrite
1 for +1 when it should not be confusing.
First of all, they arenot real numbers and donot necessarily adhere to the rules
of arithmetic for real numbers. There are times that we \act" as if they do, so we
need to be careful.
We adjoin +1 and ¡1 to R and extend the usual ordering to the set R[
f+1;¡1g. Explicitly, we will agree that ¡1 < a < +1 for every real number
a2R[f+1;¡1g. This gives the extended set with an ordering that satis¯es our
usual properties:
1) If a;b2R[f+1;¡1g, then a·b or b·a.
2) If a·b and b·a, then a=b.
3) If a·b and b·c, then a·c.
We will not extend the usual algebraic structure of the reals to R[f+1;¡1g.
Instead, when we have to we will discuss the algebra that might involve +1 and/or
¡1. Do not apply any theorem that is stated for the real numbers to the symbols
+1 or¡1.
Thesymbolsmakeitconvenienttoextendournotationaboutintervalstothereal
line.
[a;1)=fx2Rja·xg (a;1)=fx2Rja<xg
(¡1;b]=fx2Rjx·bg (¡1;b)=fx2Rjx<bg:
Occasionally, you will seeR=(¡1;1).
6.2 Sequences
Sequences are, basically, countably many numbers arranged in an ordered set that
may or may not exhibit certain patterns.
De¯nition 6.1 A sequence of real numbers is a function whose domain is a set of
the form fn2Zjn¸mg where m is usually 0 or 1. Thus, a sequence is a function
f:N ! R. Thus a sequence can be denoted by f(m), f(m + 1), f(m + 2), ....
Usually, we will denote such a sequence by fa
i
g
1
i=m
or fa
m
;a
m+1
;a
m+2
;:::g, where
a
i
=f(i). If m =1, we may use the notation fa
n
g
n2N
.
Example 6.1 The sequence f1;1=2;1=3;1=4;1=5;:::g is written as f1=ig
1
i=1
. Keep
in mind that this sequence can be thought of as an ordinary function. In this case
f(n)=1=n.
Example 6.2 Consider the sequence given by a
n
= (¡1)
n
for n¸ 0. This time we
have started the sequence with 1 and the terms look like, f1;¡1;1;¡1;1;¡1;:::g.
Note that this time the function has domainN but the range isf¡1;1g.
Example 6.3 Consider the sequence a
n
= cos
¡
¼n
3
¢
, n 2 N. The ¯rst term in the
sequence is cos
¼
3
=cos60
±
=
1
2
and the sequence looks like
1
2
;¡
1
2
;¡1;¡
1
2
;
1
2
;1;
1
2
;¡
1
2
;¡1;¡
1
2
;
1
2
;1;
1
2
;¡
1
2
;¡1;:::g:
Note that likes its predecessor, the function takes on only a ¯nite number of values,
but the sequence has an in¯nite number of elements.
Example 6.4 If a
n
=n
1=n
, n2N, the sequence is
1;
p
2;3
1=3
;4
1=4
;:::
We might use an approximation for each of these and, arbitrarily choosing 5 decimal
places, the sequence would look like
1;1:41421;1:44225;1:41421;1:37973;1:34801;1:32047;1:29684;1:27652;1:25893;:::
We would ¯nd that a
100
=1:04713 and a
10000
=1:00092.
Example 6.5 Consider the sequence b
n
=(1+
1
n
)
n
, n2N. This is the sequence
2;
µ
3
2
¶
2
;
µ
4
3
¶
3
;
µ
5
4
¶
4
;:::
or by approximation
2;2:25;2:37037;2:44141;2:48832;2:52163;2:54650;2:56578;2:58117;2:59374::::
Again a
100
=2:74081 and a
10000
=2:71815.
In looking at these examples we might think that some of them are giving us a
pattern of numbers that are \getting close" to some other real number. Others may
not give us that indication. We are interested in what the long-term behavior of the
sequence is. What happens for larger and larger values of n? Does the sequence
approach a real number? Could it approach more than one real number?
De¯nition 6.2 A sequence of real numbers is said to converge to a real number L
if for every ² > 0 there is an integer N > 0 such that if k > N then ja
k
¡Lj < ².
The number L is called the limit of the sequence.
n ¼n=2
n
1 0:5
2 0:5
3 0:375
4 0:25
5 0:15625
6 0:09375
7 0:0546875
8 0:03125
9 0:0175781
10 0:00976562
Iffa
k
g converges to L we will write lim
k!1
a
k
=L or simply
a
k
! L. If a sequence does not converge, then we say that it
diverges.
Notethatthe N inthe de¯nitiondependsonthe ² thatwe
were given. If you change the value of ² then you may have to
\recalculate" N.
Consider the sequence a
n
=
n
2
n
, n2N. Now, if we look at
the values that the sequence takes
1
2
;
2
2
2
;
3
2
3
;
4
2
4
;:::
we might think that the terms are getting smaller and smaller
so maybe the limit of this sequence would be 0. Let's take a
look and compare how N would vary as ² varies. Let's start
withsomesimplesmallnumbersandlet²be0:1,0:01,0:001,and0:0001,and0:00001.
Page 4


Sequences and Series of Real
Numbers
We often use sequences and series of numbers without thinking about it. A decimal
representation of a number is an example of a series, the bracketing of a real number
by closer and closer rational numbers gives us an example of a sequence. We want
to study these objects more closely because this conceptual framework will be used
later when we look at functions and sequences and series of functions. First, we will
take on numbers.
Sequences have an ancient history dating back at least as far as Archimedes who
used sequences and series in his \Method of Exhaustion" to compute better values of
¼ and areas of geometric ¯gures.
6.1 The Symbols +1 and ¡1
We often use the symbols +1 and ¡1 in mathematics, including courses in high
school. Weneedtocometosomeagreementaboutthesesymbols. Wewilloftenwrite
1 for +1 when it should not be confusing.
First of all, they arenot real numbers and donot necessarily adhere to the rules
of arithmetic for real numbers. There are times that we \act" as if they do, so we
need to be careful.
We adjoin +1 and ¡1 to R and extend the usual ordering to the set R[
f+1;¡1g. Explicitly, we will agree that ¡1 < a < +1 for every real number
a2R[f+1;¡1g. This gives the extended set with an ordering that satis¯es our
usual properties:
1) If a;b2R[f+1;¡1g, then a·b or b·a.
2) If a·b and b·a, then a=b.
3) If a·b and b·c, then a·c.
We will not extend the usual algebraic structure of the reals to R[f+1;¡1g.
Instead, when we have to we will discuss the algebra that might involve +1 and/or
¡1. Do not apply any theorem that is stated for the real numbers to the symbols
+1 or¡1.
Thesymbolsmakeitconvenienttoextendournotationaboutintervalstothereal
line.
[a;1)=fx2Rja·xg (a;1)=fx2Rja<xg
(¡1;b]=fx2Rjx·bg (¡1;b)=fx2Rjx<bg:
Occasionally, you will seeR=(¡1;1).
6.2 Sequences
Sequences are, basically, countably many numbers arranged in an ordered set that
may or may not exhibit certain patterns.
De¯nition 6.1 A sequence of real numbers is a function whose domain is a set of
the form fn2Zjn¸mg where m is usually 0 or 1. Thus, a sequence is a function
f:N ! R. Thus a sequence can be denoted by f(m), f(m + 1), f(m + 2), ....
Usually, we will denote such a sequence by fa
i
g
1
i=m
or fa
m
;a
m+1
;a
m+2
;:::g, where
a
i
=f(i). If m =1, we may use the notation fa
n
g
n2N
.
Example 6.1 The sequence f1;1=2;1=3;1=4;1=5;:::g is written as f1=ig
1
i=1
. Keep
in mind that this sequence can be thought of as an ordinary function. In this case
f(n)=1=n.
Example 6.2 Consider the sequence given by a
n
= (¡1)
n
for n¸ 0. This time we
have started the sequence with 1 and the terms look like, f1;¡1;1;¡1;1;¡1;:::g.
Note that this time the function has domainN but the range isf¡1;1g.
Example 6.3 Consider the sequence a
n
= cos
¡
¼n
3
¢
, n 2 N. The ¯rst term in the
sequence is cos
¼
3
=cos60
±
=
1
2
and the sequence looks like
1
2
;¡
1
2
;¡1;¡
1
2
;
1
2
;1;
1
2
;¡
1
2
;¡1;¡
1
2
;
1
2
;1;
1
2
;¡
1
2
;¡1;:::g:
Note that likes its predecessor, the function takes on only a ¯nite number of values,
but the sequence has an in¯nite number of elements.
Example 6.4 If a
n
=n
1=n
, n2N, the sequence is
1;
p
2;3
1=3
;4
1=4
;:::
We might use an approximation for each of these and, arbitrarily choosing 5 decimal
places, the sequence would look like
1;1:41421;1:44225;1:41421;1:37973;1:34801;1:32047;1:29684;1:27652;1:25893;:::
We would ¯nd that a
100
=1:04713 and a
10000
=1:00092.
Example 6.5 Consider the sequence b
n
=(1+
1
n
)
n
, n2N. This is the sequence
2;
µ
3
2
¶
2
;
µ
4
3
¶
3
;
µ
5
4
¶
4
;:::
or by approximation
2;2:25;2:37037;2:44141;2:48832;2:52163;2:54650;2:56578;2:58117;2:59374::::
Again a
100
=2:74081 and a
10000
=2:71815.
In looking at these examples we might think that some of them are giving us a
pattern of numbers that are \getting close" to some other real number. Others may
not give us that indication. We are interested in what the long-term behavior of the
sequence is. What happens for larger and larger values of n? Does the sequence
approach a real number? Could it approach more than one real number?
De¯nition 6.2 A sequence of real numbers is said to converge to a real number L
if for every ² > 0 there is an integer N > 0 such that if k > N then ja
k
¡Lj < ².
The number L is called the limit of the sequence.
n ¼n=2
n
1 0:5
2 0:5
3 0:375
4 0:25
5 0:15625
6 0:09375
7 0:0546875
8 0:03125
9 0:0175781
10 0:00976562
Iffa
k
g converges to L we will write lim
k!1
a
k
=L or simply
a
k
! L. If a sequence does not converge, then we say that it
diverges.
Notethatthe N inthe de¯nitiondependsonthe ² thatwe
were given. If you change the value of ² then you may have to
\recalculate" N.
Consider the sequence a
n
=
n
2
n
, n2N. Now, if we look at
the values that the sequence takes
1
2
;
2
2
2
;
3
2
3
;
4
2
4
;:::
we might think that the terms are getting smaller and smaller
so maybe the limit of this sequence would be 0. Let's take a
look and compare how N would vary as ² varies. Let's start
withsomesimplesmallnumbersandlet²be0:1,0:01,0:001,and0:0001,and0:00001.
For ²=0:1, we need to ¯nd an integer N so that
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:1
Lookinthetableofvalueshereandweseethatfor N =6wehavesatis¯edtheabove
condition. Following this we get the following by using a calculator or a computer
algebra system:
N > 0 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 1
N > 5 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:1
N > 9 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:01
N > 14 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:001
N > 18 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:0001
N > 22 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:00001
Wearegoingtoestablishseveralpropertiesofconvergentsequences. Manyproofs
will use a proof much like this next result. While this type of argument may not easy
to get used to, it will appear again and again, so you should try to get as familiar
with it as you can.
Theorem 6.1 (Convergent sequences are bounded) Letfa
n
g, n2N be a con-
vergent sequence. Then the sequence is bounded, and the limit is unique.
Proof: The easier property to show is that the limit is unique, so let's do that ¯rst.
Suppose the sequence has two limits, L and K. Take any ² > 0. Then there is an
integer N such that
ja
k
¡Lj<
²
2
if k >N:
Also, there is another integer N
0
such that
ja
k
¡Kj<
²
2
if k >N
0
:
Then, by the triangle inequality:
jL¡Kj<ja
k
¡Lj+ja
k
¡Kj<
²
2
+
²
2
=² if k > maxfN;N
0
g:
Page 5


Sequences and Series of Real
Numbers
We often use sequences and series of numbers without thinking about it. A decimal
representation of a number is an example of a series, the bracketing of a real number
by closer and closer rational numbers gives us an example of a sequence. We want
to study these objects more closely because this conceptual framework will be used
later when we look at functions and sequences and series of functions. First, we will
take on numbers.
Sequences have an ancient history dating back at least as far as Archimedes who
used sequences and series in his \Method of Exhaustion" to compute better values of
¼ and areas of geometric ¯gures.
6.1 The Symbols +1 and ¡1
We often use the symbols +1 and ¡1 in mathematics, including courses in high
school. Weneedtocometosomeagreementaboutthesesymbols. Wewilloftenwrite
1 for +1 when it should not be confusing.
First of all, they arenot real numbers and donot necessarily adhere to the rules
of arithmetic for real numbers. There are times that we \act" as if they do, so we
need to be careful.
We adjoin +1 and ¡1 to R and extend the usual ordering to the set R[
f+1;¡1g. Explicitly, we will agree that ¡1 < a < +1 for every real number
a2R[f+1;¡1g. This gives the extended set with an ordering that satis¯es our
usual properties:
1) If a;b2R[f+1;¡1g, then a·b or b·a.
2) If a·b and b·a, then a=b.
3) If a·b and b·c, then a·c.
We will not extend the usual algebraic structure of the reals to R[f+1;¡1g.
Instead, when we have to we will discuss the algebra that might involve +1 and/or
¡1. Do not apply any theorem that is stated for the real numbers to the symbols
+1 or¡1.
Thesymbolsmakeitconvenienttoextendournotationaboutintervalstothereal
line.
[a;1)=fx2Rja·xg (a;1)=fx2Rja<xg
(¡1;b]=fx2Rjx·bg (¡1;b)=fx2Rjx<bg:
Occasionally, you will seeR=(¡1;1).
6.2 Sequences
Sequences are, basically, countably many numbers arranged in an ordered set that
may or may not exhibit certain patterns.
De¯nition 6.1 A sequence of real numbers is a function whose domain is a set of
the form fn2Zjn¸mg where m is usually 0 or 1. Thus, a sequence is a function
f:N ! R. Thus a sequence can be denoted by f(m), f(m + 1), f(m + 2), ....
Usually, we will denote such a sequence by fa
i
g
1
i=m
or fa
m
;a
m+1
;a
m+2
;:::g, where
a
i
=f(i). If m =1, we may use the notation fa
n
g
n2N
.
Example 6.1 The sequence f1;1=2;1=3;1=4;1=5;:::g is written as f1=ig
1
i=1
. Keep
in mind that this sequence can be thought of as an ordinary function. In this case
f(n)=1=n.
Example 6.2 Consider the sequence given by a
n
= (¡1)
n
for n¸ 0. This time we
have started the sequence with 1 and the terms look like, f1;¡1;1;¡1;1;¡1;:::g.
Note that this time the function has domainN but the range isf¡1;1g.
Example 6.3 Consider the sequence a
n
= cos
¡
¼n
3
¢
, n 2 N. The ¯rst term in the
sequence is cos
¼
3
=cos60
±
=
1
2
and the sequence looks like
1
2
;¡
1
2
;¡1;¡
1
2
;
1
2
;1;
1
2
;¡
1
2
;¡1;¡
1
2
;
1
2
;1;
1
2
;¡
1
2
;¡1;:::g:
Note that likes its predecessor, the function takes on only a ¯nite number of values,
but the sequence has an in¯nite number of elements.
Example 6.4 If a
n
=n
1=n
, n2N, the sequence is
1;
p
2;3
1=3
;4
1=4
;:::
We might use an approximation for each of these and, arbitrarily choosing 5 decimal
places, the sequence would look like
1;1:41421;1:44225;1:41421;1:37973;1:34801;1:32047;1:29684;1:27652;1:25893;:::
We would ¯nd that a
100
=1:04713 and a
10000
=1:00092.
Example 6.5 Consider the sequence b
n
=(1+
1
n
)
n
, n2N. This is the sequence
2;
µ
3
2
¶
2
;
µ
4
3
¶
3
;
µ
5
4
¶
4
;:::
or by approximation
2;2:25;2:37037;2:44141;2:48832;2:52163;2:54650;2:56578;2:58117;2:59374::::
Again a
100
=2:74081 and a
10000
=2:71815.
In looking at these examples we might think that some of them are giving us a
pattern of numbers that are \getting close" to some other real number. Others may
not give us that indication. We are interested in what the long-term behavior of the
sequence is. What happens for larger and larger values of n? Does the sequence
approach a real number? Could it approach more than one real number?
De¯nition 6.2 A sequence of real numbers is said to converge to a real number L
if for every ² > 0 there is an integer N > 0 such that if k > N then ja
k
¡Lj < ².
The number L is called the limit of the sequence.
n ¼n=2
n
1 0:5
2 0:5
3 0:375
4 0:25
5 0:15625
6 0:09375
7 0:0546875
8 0:03125
9 0:0175781
10 0:00976562
Iffa
k
g converges to L we will write lim
k!1
a
k
=L or simply
a
k
! L. If a sequence does not converge, then we say that it
diverges.
Notethatthe N inthe de¯nitiondependsonthe ² thatwe
were given. If you change the value of ² then you may have to
\recalculate" N.
Consider the sequence a
n
=
n
2
n
, n2N. Now, if we look at
the values that the sequence takes
1
2
;
2
2
2
;
3
2
3
;
4
2
4
;:::
we might think that the terms are getting smaller and smaller
so maybe the limit of this sequence would be 0. Let's take a
look and compare how N would vary as ² varies. Let's start
withsomesimplesmallnumbersandlet²be0:1,0:01,0:001,and0:0001,and0:00001.
For ²=0:1, we need to ¯nd an integer N so that
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:1
Lookinthetableofvalueshereandweseethatfor N =6wehavesatis¯edtheabove
condition. Following this we get the following by using a calculator or a computer
algebra system:
N > 0 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 1
N > 5 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:1
N > 9 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:01
N > 14 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:001
N > 18 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:0001
N > 22 implies
¯
¯
¯
¯
N
2
N
¡0
¯
¯
¯
¯
< 0:00001
Wearegoingtoestablishseveralpropertiesofconvergentsequences. Manyproofs
will use a proof much like this next result. While this type of argument may not easy
to get used to, it will appear again and again, so you should try to get as familiar
with it as you can.
Theorem 6.1 (Convergent sequences are bounded) Letfa
n
g, n2N be a con-
vergent sequence. Then the sequence is bounded, and the limit is unique.
Proof: The easier property to show is that the limit is unique, so let's do that ¯rst.
Suppose the sequence has two limits, L and K. Take any ² > 0. Then there is an
integer N such that
ja
k
¡Lj<
²
2
if k >N:
Also, there is another integer N
0
such that
ja
k
¡Kj<
²
2
if k >N
0
:
Then, by the triangle inequality:
jL¡Kj<ja
k
¡Lj+ja
k
¡Kj<
²
2
+
²
2
=² if k > maxfN;N
0
g:
Therefore jL¡Kj < ² for any ² > 0. But the only way that that can happen is for
L=K, so that the limit is indeed unique.
Next, we need to prove boundedness. Since the sequence converges, we can take
any ² we wish, and tradition shows us to take ² = 1. Then there is an integer N so
that
ja
k
¡Lj< 1 if k >N:
Fix that integer N. Then we have that
ja
n
j·ja
n
¡Lj+jLj< 1+jLj=P for all n>N:
Now, de¯ne M = maxffja
k
j;k = 1;:::;Ng;Pg. Then ja
n
j < M for all n, which
makes the sequence bounded.
6.3 The Algebra of Convergent Sequences
This section proves some basic results that do not come as a surprise to the student.
Theorem 6.2 If the sequence fa
n
g converges to L and c 2 R, then the sequence
fca
n
g converges to cL; i.e., lim
n!1
ca
n
=c lim
n!1
a
n
.
Proof: Let's assume that c6=0, since the result is trivial if c =0. Let ²> 0. Since
fa
n
g converges to L, we know that there is an N 2N so that if n>N
ja
n
¡Lj<
²
jcj
:
Thus, for n>N we then have that
jca
n
¡cLj=jcjja
n
¡Lj<jcj
²
jcj
=²:
which is what we needed to prove.
Theorem 6.3 If the sequence fa
n
g converges to L and fb
n
g converges to M, then
the sequence fa
n
+b
n
g converges to L+M; i.e., lim
n!1
(a
n
+b
n
)= lim
n!1
a
n
+ lim
n!1
b
n
.
Proof: Let ²> 0. We need to ¯nd an N 2N so that if n>N
j(a
n
+b
n
)¡(L+M)j<²:
Sincefa
n
g andfb
n
g are convergent, for the given ² there are integers N
1
;N
2
2N so
that
If n>N
1
thenja
n
¡Lj<
²
2
and if n>N
2
thenjb
n
¡Mj<
²
2
Thus, if n> maxfN
1
;N
2
g then
j(a
n
+b
n
)¡(L+M)j·ja
n
¡Lj+jb
n
¡Mj<
²
2
+
²
2
=²: