REAL
ANALYSIS
PREP
THE WELL-ORDERING PRINCIPLE
well
the
AXIOM
[ THE WELL
2
Every
non
THEOREM
For
[ THE
3
K
each
the
on
Pln )
Proof
52
↳
:
1 ( 2
Oc
↳
9
i.
S2
go
i.
9
,
C- S
But
0cg ,
↳ However
5291
↳
:
PCKI
there
is
true
is
some
way
for
all
of
KEN
depending
that
showing
.
:
[ Well
T2
<
I
IN
C- Z
Eq
Ordering ]
}
go
≠
∅
.
( 2
go
↳ Contradicts
5290
so
,
5290
=
,
Ll
(
0C 91
↳ But
Therefore
.
IN
true / false
is
Z
c-
element
i.
I
-
C-
{q
s=
→
4
C-
Pllc)
152-1190
=
,
[
true
n
(
C- IN
9
.
smallest
a
↳
:
is
each
K
number
contradiction ]
by
pig
0
>
for
and
the
concerning
number
Pig
rational
is
has
s
↳
q
52
↳ :
[ proof
:
proposition
Plnti )
rational
a
.
INDUCTION ]
true
is
element
logical
a
that
implies
=
Assume
↳
be
Pll )
rational
is
.
smallest
a
MATHEMATICAL
PCKI
not
:
has
OF
If
.
is
52
↳
K)
2
:
let
,
true
:
if
:
IN
of
being
THEOREM
PRINCIPLE
C-
value
1N
of
subset
empty
-
PRINCIPLE ]
ORDERING
-
principle
sanderling
C- Z
Z
C-
go
-
i.
be C- IN
must
91
cqo
290-5290
5215290 -90 ]
=
and
91<90
=
go
the
is
smallest
.
!
9,
C- S
of 5
element
NUMBER SYSTEMS
IN
-
Z
-
④
-
IR
-
natural numbers
{
the
{
integers
rational
{
numbers
real
{
numbers
Algebraically
,
it
1,2
.
.
.
,
-3 , -2 , -1
mln
-
3,4
,
x.ro
:
.
}
.
0,112,3 }
,
MEZ
IN }
in C-
}
R
distinguish
to
difficult
is
.
,
and
02
.
Thus
we
need
another
axiom
.
Natural numbers
has
↳
Every
↳
Natural
numbers
are
↳
They
bounded
below
A
are
subset
We
i.
↳
subset
Any
,
call
number
A
≤
m
smaller
smallest
a
"
separated
n
:
IR
a
Mtn
for
all
"
bound
M
"
lm.nl
=)
n
C-
for
is
≥
I
IN
it
below
bounded
lower
CW-0.PT
element
:
so
is
than
"
there
exists
A.
also
a
lower bound
.
an
M
C- IR
sit
.
m≤ x
for
all
✗
C-
A
THE
COMPLETENESS
THE
COMPLETENESS AXIOM
Any
①
m
②
For
THE
≤
for
each
Any
A
non
IR
c-
We
It
A
M
is
bounded
MAXIMUM
[ definition]
①
for
every
②
b
IA )
①
for
→
sup (A)
A
*
A
*
It
*
It
(A)
/ A)
②
b
*
C-
*
It
* It
has
above
exists
number
a
least
a
that
such
m
bound
upper
x
x
KEA
,
}
bounded
is
below
.
the
of
maximum
the set
A
of
A
it
:
A
the
is
least
bound
upper
it
:
C- A
a
b
then
,
≤
C.
.
maximum
a
C-
A
it
always
it
is
has
a
does
sap (A)
,
bounded
is
[ even
.
above
/ A)
Max
We
C- A
,
above :
supremum
not
SUPIA )
above]
bounded
=
need
be
to
Max
.
an
element
of
A
of
A
.
1A )
b
say
a
the
is
of
minimum
the set
A
of
A
it
:
b
≥
a
a
=
A
C-
for
int
(A)
exists
min
,
greatest
lower bound
it
:
b
every
a
C- A
minimum
a
bounded
is
exists ,
the
is
,
b ≥ c.
then
'
have
not
b
that
say
a ≥
,
c
which
Min / A)
min / A)
we
:
≥
need
set
of A
maximum
every
infimum
A
§
-
is
every
bounded
,
A
conditions :
.
b
have
:
a
=
whenever
*
A
is
[ definition]
set
Cn
x
A
mail.AT
A
with
INFIMUM
t
every
→
of
exists ,
for
②
EA
x
A.
=
b
that
for
exists
①
for
an
b
≤
≤
c
which
[ definition]
①
a
not
Minimum
Infimum
b
say
say
,
.
two
of A
supremum
MINIMUM
→
A
≤
a
set
Max
satisfy
supremum
a
We
:
C-
a
need
Max
must
m
bound
infimum
=
,
maximum
=
set
lower
A
every
whenever
number
a
bounded
is
for
A
-
=
C- A
[ definition]
②
is
there
it
→
We
:
a
=
Supremum
above
bound
Maximum
→ Max
,
greatest
SUPREMUM
AND
C-
above
bound
upper
there
,
that
IR
upper bound
an
lower
least
men
of
bounded
call
a
a
2 :
subset
is
Greatest
*
AXIOM
empty
-
with
n
COMPLETENESS
A
set
has
C- A
sc
number
below
bounded
is
for
bound
lower
all
that
IR
of
-
x
IR
OF
Ii
empty subset
the
greatest
non
To be
•
AXIOM
is
below
(A)
C-
A
,
[ even
it
it
is
always
has
a
int
does
bounded above :
CA )
int (A)
below ]
bounded
infimum
not
=
need
to
min .LA )
be
an
element
.
.
≤
m
for
all
✗
C- A
.
BOUNDED
A
SUBSETS
subset
A
bounded
is
Example
1-
:
IR
of
A
iff
there
.
{ sin ≥
=
is
number
a
KEIR }
:
it it
bounded
is
where
both
bounded
is
B
that
such
:
above
1×1<13
for
and
all
below
x
C- A
B=I
CONSEQUENCES OF THE
COMPLETENESS AXIOM
THE
EXISTENCE
let
A
{
=
A- =/ ☒
↳
-
IR
c-
x
-
-
-
-
x2
:
y
-
y
}
C2
A
C-
0
since
-
-
-
TE
OF
__
2
1
I
•
•
42
A
Claim
The number
:
Proof : ① Consider
i.
Proof
:
①
s
③
It
any
0
first
h
<
hcl
Csth )2
④ By
choosing
But
's
that
that
us
A
A
and
≤ 2
has
least
a
sup (A)
=
bound
upper
2
<
SZCZ
Csth / 2
20 ,
i.
then
tells
x
>2
a
&
A
52
Suppose
② For
Thus
,
Axiom
0
with
0C
for
bound
upper
an
74
> 2x
that
=
is
number
Completeness
such
;
2
any
x2
② The
Claim
:
,
hzch
then
S2 1-
(
h
5th C- A
=
S2
,
tzsh
so
th
Csthl ?
?
L
S2
1-
[ 5=2 ]
4h th
5h
sufficiently
:S
small
isn't
,
an
we
can
upper
make
s2
bound
for
1-
5h
A
22
,
S
.
.
Now
①
:
52
Suppose
72
h>0
② For any
i. ( s HP
then
This
④
52
Let
a.
Then
s
bi
real
{
=
]
[ 92
≥
I
Nested
9
Interval
tells
↳ The
↳ The
Put
↳
bounded
↳
↳
is
A
has
since
A
Each
is
bk
set
The
Archimedean
:
① Thus
≤
s
there
-
③ Contradiction
:
-
.
nested
is
.
it
:
.
of
sequence
have
"
any
closed
holes "
and
bounded
intervals
.
endpoints ]
two
n C-
s
S
for
bk
s ≤
i.
,
bk
has
For
:
biggest
no
each
with
x
≤
n
C-
IR
element
there
,
exists
all
for
x
.
n
n
C- IN
IN
C-
abovebyx
say
IN
A
K C- IN
all
for
s
≤
9k
,
Axiom ]
PROPERTY
IR
-
[ Completeness
:S
Caib ]
Axiom
,
is
.
.
}
.
.
≤
s
Property
( S
l
.
numbers
I C-
Completeness
[
.
bound
bounded
is
.
.
bi
nested
bound
upper
natural
Suppose
IN
Since
Thus
of
[ 93 .bz ]
interval
.
bound
ARCHIMEDEAN
The
≥
bounded
and
"
£2
doesn't
92,93
SE
Proof
②
,
upper
an
i.
The
,
closed
a
bounded
upper
ak
IHE
bound
upper
bi
by
is
Any
closed
be
{9
=
an
i.
-
0
>
b
[ 93,53 ]
≥
lbs
nested
be
least
a
5
be
must
intervals
Proof :
A
least
[ 92 .bz ]
,
bz ]
:
is
[ 9 ,b , ]
,
IR
must
must
intervals
x
≤
a
≤ b]
! 43
that
us
intervals
↳ The
h
of
values
the
with
≤
a
:
,
Property
intersection
↳ This
b2
small
not
is
intervals
9.
The
all
numbers
KEIR
of
sequence
,
1-
IR
be
b
[ ai
Zsb
INTERVAL PROPERTY
[ a ,b ]
A
that
C-
THE NESTED
for
32
-
-
4h
-
CS b)2
means
i.
S2
S2
=
-
>
-
③ But
CS b) 2
,
is
1
not
with
must
s
be
has
IN
that
an
-
i
a
greater
and
than
bound
upper
bound
upper
< n
least
then
all
1N
for
s
<
n
ntl
f IN
.
,
S
.
with
x
en
.
has
a
non
-
empty
two
If
If
≠ bi
a
Where
The
numbers
real
x
-
Archimedean
Thus
IR
¥,
then
its
=
sit
IR
c-
la
.
-
bl
<
for
tn
all
C-
n
then
IN
a
=
b
.
i
at
Property
not
does
,b
a
contain
there
that
says
infinitesimals
any
fintinite.ly
small
Tn
with
IN
C-
n
an
is
<
¥
numbers ]
CARDINALITY
A
A
We
B
onto
We
have
B
and
Sets
it
as
A
:
-
( size )
if
there
is
bijection
a
( one
-
to
B
to
cardinality
a
used
comparison
in
①
{ 11213
②
IN
size
cardinality
same
.
write
assign
the
ii.
,
classes
are
comparing
to
it
a
well
known
set
.
:
}
in
by
set
a
for
follows
as
n
some
C- IN
:
All Sets
Infinite Sets
Finite
Set
Uncountable sets
countable sets
A
set
↳
finite
↳
Infinite
An
is
it
infinite
↳
uncountable
i. We
A
A
is
A
it
it
A
i.
,
i.
in
for
}
some
[ has
C- IN
n
n
elements ]
finite
not
:
has
write
can
1,2
is
countable
is
{
~
A
set
countable
Useful
A
if
↳
A
:
the
not
is
iff
the
cardinality
same
as
IN
countable
there
.
is
d-
elements
a
f
bijection
A
as
be
written
:
infinite
an
IN
→
sequence
A
.
.
results :
set
is
subset of
countable
a
ift
countable
it
can
set
is
either
as
an
finite
infinite
or
sequence
countable
.
Or
A
=
&
-
one
correspondence )
from
Claim
①
C2
:
we have
Like
to
②
¥'
E
&
i
I
to
¥
¥
E.
}
¥
%
¥
¥
%
:/
÷
÷
±
÷
5-
¢
:
:
:
:
:
a
IR
:
by
{
:
1
2
,
sequence
a
1)
,
We
I
✗2
V3
✗4
✗
5
=
0
number
in
=
0
bi
:
b
•
o
↳ Contradiction
b
found
.
.
.
.
.
.
3
..
4
,
,
{
}
,
¥
,
É
,
-
}
-
_
,
.
.
.
24
.
=
=
=
{
-
.
_
.
.
.
-
b
74.x
i.
its
in
with
,
decimal
}
I n =2
=/
}
✗ 22=12
✗ 22=2
it
2
3
×
representation
,
b
xnn≠2
it
seek
b
a
i.IR
is
.
form :
decimal
xnn=2
≠ Az
}
b
≠
In
C- [ 0,17
≠
}
=
711=12
it
3
✗
-
it
2
sequence
.
.
254
it
3
=
infinite
.
it
2
countable
sequence
245
.
countable
is
}
number
blbzb ≥
.
=
bn
i.
have
_
the
742244
b3
↳ we
.
,
✗ 32×33734
be
and
:
.
-
an
12713714
real
a
bz
Then
.
-
is
222×23 ✗
✗ 31
.
b.
follows
✗
)
0-251252×53
=
.
.
.
IR
is
V3
0.241
=
.
,
,
,
,
0.221
=
construct
as
142
0,711
=
Co
there
71
each
write
7C
{
=
.
É
,
Suppose
that
means
0
.
6
¥
,
i.
(
.
.
uncountable
is
Contradiction :
This
.
.
§
Proof
as
'
z
Claim
Now
É
4-
Sequence
_
Q
of
elements
¥
}
I
:
"
the
write
:
¥
-
countable
is
for
E
all
Cosi)
uncountable
K
C-
$
.
IN
{
× ,
,Xz
.
.
.
}
:
.
.
.
Sit
:
SEQUENCES
sequences
LIMITS
SEQUENCES
OF
→ Consider
We
N
a
C-
Interval
:
after
which
:
( l
Consider
the
each
lies
in
{ y c-
=
( UK
for
it
always
lte )
,
1<=1
l
=
ak
e
-
CONVERGENCE
lak )
:
an
( 3- NEIN )
)
> o
Example
tim
here
IN
( He
sequence
that
say
1-
IN )
C-
112
K ≥
9k
sequence
interval
the
ly
:
-
interval
et
3k
+ a
,
5k -6
we
that
guess
Here
i.
tak
,
i.
-
this
bl
I
=
¥b
l
to
converges
=
K
late
or
.
b.
around
,
-
there
It
exists
number
a
< E
}
-
ll
CE
C- IN
}
38
3- /
-
ce
late
IN
=
{ so
,
=
K
it
s-cs-k.co )
> I
38
i.
<
E
51C
-6
[ 5k b)
5
-
31
<
5E
3¥
i.
(
tb
5k
,
i
3¥
¥
+
,
Given
E.
any
so
1k
,
Archimedean
the
31
¥
+
252
K
i.
Claim
Proof
:
:
≥
late
IN
lim
3k to
Kyo
51=-6
Given
any
=
E. so
-
el
L
Property
L
we
says
find
can
N
E
¥
,
NEIN
choose
¥7
+
¥
< N
so
that
N
> I
and
:
an
N
C-
IN
:
For
19k
i.
-
l/
K
number
natural
any
1314-0
¥-0
=
≥N
§/
-
38
=
38
{
515k b)
515N b)
-
38
=
5
A
Proof
①
him
②
dim
Suppose
:
l
9k=l
( V-E
means
9k=m
so
( V-E
means
so
limit
one
both
are
m
) CFP
C- IN )
.
limits
K ≥P
( V-kc.IN )
K ≥Q
=)
,
19k
=)
19k
-
-
m
l/
42
<
/
<
42
↳✗
Starting
For
with
KE
any
E
any
IN
> ◦
find
i
P
K
≥
talc
-
and
P
l /
Il MI
talc
≤
Il MI
-
<
E
then
Calc )•
calc )
is
is
Cak )
If
with
for
talc
-
let
all
-
l
,
aktml
F added
f
late MI
t
-
Elz
f
E >0
-
-
Archimedes
Cakl
bounded
C9k1=
any
sequence
bounded
above
it
CFM
bounded
below
it
( Fm
bounded
is
is
Max
ak
9k
{ AQ }
=
92
<
late ml
it
bounded
sequence
,
need
t.FM ,m
we
C- IR )
can
not
,
ltfk
/
l
-
m
-1
✗
(l
/
◦
=
l=m
its
is
c- IR )
C- IN )
lfk
,
m
,
however
,
the
,
≤
ale
≤ M
ak
≤
ak
,
EIN )
i¥µ 10kt
consider
{ ate
of
set
LUKE IN )
GIR)
converge
and
triangle inequality
says
,
-
E
i.
Associated
N=
put
K ≥ Q
and
Elz
<
Since
342
(
=
-
i.
and
IN
Q C-
,
:
N
Ic ≥
A
C9kl=
of
( V-kc.IN )
C- IN )
) CFQ
E
=
381g
than
more
and
38_
{
(56-0)
have
can't
sequence
-
ik
C- IN }
M
≥ m
.
{YIN
lakl
converse
isn't
true
.
-
m
)
Claim
Every
:
Proof
( He
let
Put
m
ltk
RULES
C-
IN
:
i
NEIN
§
9
,
{
9
,
:
s.li
92
,
≤
m
Example
za
!i→%
IN )
C-
19k
.
.
'
.
19N
'
.
.
/
.
K ≥
late
IN
-
ll
e
E
<
1
for
natural
any
K
number
≥
N
}
}
I
≤ M
ak
LIMITS
WITH
}
,l
-
ll
-
ltl
,
AN
can ?=
Suppose
:
=
,
.
192
WORKING
FOR
( UK
,
mm
=
Introductory
R.I.P
=
Max
=
bounded
is
Ak =L
( 3- NEIN )
)
exists
M
sequence
Kyra
so
E.
There
i.
lim
Suppose
:
i.
i.
convergent
from
directly
is
,
definition
the
¥69k
with
sequence
a
-_
2
.
-
P:
( He
Ii
so
( 3- NEIN )
)
1T¥
-31
,
39k
IN )
C-
K ≥
late
IN
et
CE
L
E
-
ce
( 291C
2
-
( UK
-
1)
30k
-
49k
1-2
=
Gale -3
60k
-3
2- 9k
=
Gale
-3
19k -21
=
3129k -11
For
a
values
large
22129k
:
il
lai
i.
3 / Zak
i.
-
of
Given
for
Also
E
an
all
choose
1C ≥
,
.
an
C
/
/ Clk
-21
is
"
small
"
-11=3
while
129k
an
that
19k -21
E
(
3×2
choose
,
talc -21
1
4
4
20
N
K
N
'
C-
IN
so
L
GE
'
N
"
C-
IN
so
that
late -21
C
'
z
for
all
lc
≥
N
"
For
K
such
Now
N=
put
For
all
IN
-
4
<
I
{ N' IN " }
Max
C-
K
29k
C
2
:
K ≥ N
,
talc -21
/ Zak
3
-
3129k
let
:
Cak)
Then
Cbk )
and
tim
( te
> 0
)
( FN
19kt
all
At
the
> 0
time
same
numbers
N
we
,
¥7s
.
Max
=
K
IN
≥
ak
=
i
Hitfix
and
9
E
L
/
bk
=b
K
≥
{
N' IN
=)
N
IN )
-
%?
=
K
9k
/
≥ IN
b) I
L
N
an
%:
t
laktbk )
'
EIN
s
.
-1
find
C- IN
N"
sit
119k
≤
.
lbk
.
}
≤
-
Catbl I
L
E
E
"
"
bk
19k
-91
<
bl
<
Elz
42
for
'
N
can
we
bk )
find
can
to
,
C-
a) tlbk
numbers
natural
natural
Put
E
any
-
( Fk
)
C- IN
119k
i.
Given
s.li
-
:
Kero
Proo
be
3×2
talc -21
i.
RULE
bE_
<
,
,
-
tak
a)
-
Elz
+
Cable
al
t
t
-
bl
Ibk
Elz
-
bl
=
E
-
for
all
RULE
let
:
Cak)
Cbk )
and
be
s.li
.
nx
ak
and
9
=
I
!↑xbk=b
Then :
Iim
Cakbk )
↳✗
( V-E
Proo
Rough
> 0
lakbk
:
)
( 3- NEIN
abl
-
) ( Hk
I
=
tak
-
E
=L
we
,
For such
find
can
K
E. 20
Choose
N
Put
i.
"
s.tl
C- IN
K
N
choose
,
N
-
lblcl
i.
Given
Ibl
-
,
=
≥
Max
N
{
=)
<
"
ble
N
≤
I
'
,
C-
sit
b- I
Ibl
IN
is -1
.
bl
N"
1-
' "
lbk
bk
E
lbk
-
bl
<
bl
1
btl
2
all
for
Ibl
≤
1C
2N
'
1- I
<
zc÷+ ,
for
,
for
all
lc
≥
N
all
K
≥ N
" '
}
fate
<
at
-
<
612
19k -91
.
Ibl 1- 11
lbk
191
abl
-
1
/ blclc
-
-
)
abt
-
t
Etclalti )
C
,N
<
able
t
bk
lakbk
=)
+
able
al lblcl
EIN
'
N
-
K ≥ N
Elz
≤
If
IN )
C-
akbk
≤
( ¥19k) / %:
=
bl
2%11-11
<
e-
21141-1 )
lakbk
-
abl
≤
≤
talc
-
al
lbkl
e-
t
"b / ᵗ '
)
la / lbk
19 /
+
2C / bit , ,
c
lbltl
§
t
1-91
191+1
↓
21
<
Elz
t
Elz
=
E
-
bl
2%+1 ,
§
"
THE MONOTONE
CONVERGENCE THEOREM
Recap
↳
A
↳
If
:
bounded
sequences
Definition
↳
↳
A
THM
Cak )
19kt :-. ,
,
53
is
is
THE
:
its
:
eventually
decreasing
,
does
bounciness
then
imply
convergence
.
is
iff
either
is
CONVERGENCE
MONOTONE
both
is
IN
C-
.
.
decreasing
or
=)
IN
C-
K
increasing
n
m ≤
.
all
for
K
for all
akti
≤
akti
ak ≥
that
one
9k
for
an
≤
am
all
C-
min
IN
THEOREM
bounded
and
increasing
above
then
,
converges
it
supremum
Cale )=
We
want
to
s
This
means
Because
Thus
ak
S
lak
sequence
theorem
-
Sl
that
helps
-
<
E
L
is
an
ak
E
≤
both
≤
for
s
that
a
-
K
all
for
given
,
:
S
<
an
E
≥
N
s
sup {
=
9k
,
KEIN }
is
K
not
C-
IN
upper bound
an
.
.
K
all
E
-
:
.
all
for
s
set
S
caste
decreasing
≤
this
with
IN
ale
be
ak
:
,
supremum
a
E >o
for
required
as
prove
NG
an
,
ale
bound
upper
increasing
is
for
bound
find
can
we
:
i.
least
the
let
:
has
it
then
,
¥?xak=s
upper
an
is
above
that
show
is
s
bounded
is
,
'
.
Every
it
increasing
If
-
it
.
Because
This
.
or
increasing
decreasing
sequence
a
to
increasing
is
,
sequence
sequence
↳ If
Proof
k=
monotone
.
convergent
:
Similarly
A
be
not
eventually
are
sequence
↳ A
need
sequence
3- N
.
and
limit
bounded
exists
below
without
to
converges
having
to
find
its
its
infimum
value
.
.
SERIES
Definition
↳ The
↳
partial
We
of
sums
Éak
£9k
series
a
k=
a
say
If
↳
:
the
that
say
converges
,
then
are
,
9,1-92
,
it
converges
series
define
we
9
it
911-921-93
,
I
its
be
s
.
Iim
=
NSX
↳ If
the
If
different
two
¥9k
=
É=
PROPOSITION
bk
,
56
though
we
even
Note
:
É=
do
have
not
him
then
,
9k
ntl
Let
s
sum
a
have
ak
=
for
15-1
.
.
the
bk
converges
[ a,
divergent
is
series
sums
same
all
sum ,
C- IN
K
we
can
write
.
=
◦
§=
=
h
9k
-
for
29k
,
¥
n→x
9k
C-
IN
.
n
ntl
Iim
=
,
"
S
=
=
Irm
2
ng
nsrx
i.
n
,
limant
-
all
lC=1
n
lim
=
'
<
the
partial
n
that
anti
i.
then
Ébk
and
9k
,
,
have
doesn't
it
series
converges
k=l
Proof
then
,
diverges
sums
.
:
£9k
↳ If
partial
diverges
series
a
If
Ot
sequence
_
of
sequence
to
sum
,
ak
-
=
ale
'
-
lim
£-19
ng
"
S
0
0
=
KS
PROPOSITION
↳ If
a
series
58
:
N
{
k=0
ale
is
convergent
,
then
so
is
the
¥59k
series
with
[
calc
k=0
PROPOSITION
If
two
Éo
2
59
c
any
constant
Eak
10-0
:
sequences
( aktbk )
=
for
ale
and E
K=O
bk
are
convergent
,
then
so
is
the
series
C
.
Thus
{
:
Chak
+
Sbk)
is
Definition
a
geometric
some
series
62
air
Irl
and
Eble
convergent
are
is
-
,
is
of
{
form
the
Ark
for
k=0
:
C- IR
< 1
it
it
series
EIR
air
Proposition
If
Eak
61
An infinite
let
it
convergent
Kao
with
then
=/ O.lt
a
it
In
with
converges
1
≥
then
,
geometric
the
%
series
Ark
diverges
,
.
:
✗
[ ark
a
=
k=0
Proof
The sequence
:
this
Irl ≥
Now
.
Then
to
it
r=1
it
0
=
divergent
is
Irl
Sn
Sn
(
it
r
Ar ?
t
ar
a
=
) Sn
Itr
=/ 1
=
rsn
-
art
tart
a
=
rsn
Lt
.
THEOREM
then
all
=
rntl
-
rnt
-
series
the
64C THE
Series
proving
Earle
series
sn
for
Sark
=
r
all
=
and
-1
it
diverges
C-
n
IN
tarn
.
.
.
For
that
E
each
-
tqrnt
'
)
r
-
lim
=
with
convergent
is
'
)
'
Sn
Ye
l
Csn )P= ,
=
=
IN
SERIES )
It §
It
=
C-
n
52N
HARMONIC
,
let
sn
bounded
is
¥
1-
•
=
r
divergent
is
.
É :-c
that
Note
-
:
.
.
:
a
=
ns
:
Put
.
it
Or
,
Earl
Proof
the
> 1
arntt
-
l
11-4121
.
1-9-37
all
=
Sn
The
Thus
.
Irl
it
Koo
:
Thus
r
-40
a
rk
Iim
that
note
-
Cork )
terms
converges
sequence
1
of
l
.
We will
for
N
C-
the
that
prove
each
.
IN
is
series
divergent
by
:
[ IT
It
( the
{
t
last
C 's
1-
term
to
1-
in
It's )t ( ta
each
1-
bracket
to
1-
IT
is
1-
a
¥
1-
%
power
1-
Fat's
Ot
t
2)
%)
t.n.tl
-
"
1-
ÉN )
,
≥
I
=
for
that
each
Csn)
ᵗz
,
we
unbounded
is
tz
t
¥
t
%
t
{
f-
+
t
£
.
t
.
.
2m¥
t
.
.
.
t
É
ÉN
EIR
M
TE
t
t
It
=
Thus
É
If
find
can
NEW
with
≥
Szn
I
1-
IN
>
M
This proves
.
.
THE COMPARISON TEST
THEOREM
let
{ bic
£9k
If
•
:
COMPARISON
THE
Cak)
It
•
65
convergent
divergent
is
FOR
be
Cbk)
and
is
TEST
then
,
for
ble
≤
9k
all
K C- IN
:
{ bk
is
so
≤
£9k
is
so
0
that
such
then
,
SERIES
Note :
① It sufficient that
②
It
③
Only
to
the
have
that
series
Zak
series
non
-
negative
terms
.
{ ble
but
converges
K
at
values
diverges
.
:
Since
•
tha
applies
PROOF
large
N
possible
is
for
hold
inequalities
the
9k
≥
and
0
ble
≥
for
0
all
K
,
the
of
sequences
partial
sums
({ak )
n
( ◦bk )
and
•
•
Ébk
If
Let
M
Since
•
0
both
are
is
convergent
C- IR
be
≤
ate
≤ ble
leak )
%
◦
•
•
all
The
If
so
Monotone
{ ak
{ ble
n
C- IN
is
also
can't
bound
upper
for
all
≤
Ébk
K
for
C- IN
≤
and
CÉBK )
bounded
i.
is
above
.
.
:
m
.
bounded above
Convergence
diverges
.
( { bk )
is
so
,
an
Éak
for
increasing
,
.
Theorem
partial
sum
converge
.
53
is
that
says
unbounded
.
:
( Éak )
partial
sums
convergent
is
Ot
ble
is
unbounded
THEOREM
let
The
-
SERIES
constant
converges
•
≤
p
all
For
For
Ep
is
1
application
simple
a
1C
p
1
>
each
C-
,
IN
the
C-
,
we
the
of
find
.
It
p
,
we
'
to
only
C- IN
is
n
≤
'
2N
-1
C
Ep
It
=
( Ip )
É
t
p
°
( Ip )
( typ
t
-
-
-
+
t
p
t
Fp
t
C :-p )
Ip
t
'
1-
+
( Ip )
?
Épt Tp
+
)
Ip )
:
geometric
t
( Ip / 3
Ip
<
1
series
≤
@ Ep )
i.
Converges
É
I :-p )
is
bounded
is
"
both
≤
{
(Ep )
"
increasing
and
t
.
-
.
.
Ip
"
convergent
n
Monotone
:
↓
E ,÷p
the
increasing ,
CE / KP )
that
show
with
ltzptzp )
≤
=
:
> 1
have
diverges
@ / KP )
of
N
£11k
series
need
we
It
1-
i.
1
test
comparison
É÷p
=
.
≤
p
/ KP
sums
can
≤
'
<
Harmonic
that
says
IN
1k
the
partial
Theorem
n
'
C
and
ÉÉp
•
it
diverges
,
:
Convergence
•
&
series
.
O
for
The
.
:
case
that
p
a
2 ,÷p
PROOF
THE
i
be
p
then
•
71
bounded
above
+
.
.
.
1-
( Ip )
""
above
.
,
THE BOLZANOWEIERSTRASS THEOREM
* THIS
ABOUT
IS
DEFINITION
let
72
can)
be
increasing
a
to
the
PROOF
numbers
.
Then
Chief
suppose
came)
is
is
of
subsequence
a
strictly
a
can)
converges
then
,
ot
subsequence
every
can)
converges
:
lnigmx
/
an
lank
-
is
Iim
1
for
means
for
E
E.
≥
n
N
increasing
d-
subsequence
nk
each
all
strictly
a
9
≤
for all
lcf IN
has
least
so
,
find
can
we
an
N
C- IN
.
of
sequence
natural
numbers
,
so
Can )ʳ
.
It follows
it
that
K
≥
N
i
then
me
≥N
E.
<
clinic
this
<
be
K
a
,
at
-
Cmc)
that
i.
a
=
Canio )•
Notice
so
an
that
that
•
and
numbers
limit
same
Now let
•
can ]
sequence
such
•
natural
of
real
74
Suppose
•
of
sequence
a
sequence
PROPOSITION
If
*
SEQUENCES
9
=
KS
COROLLARY
If
75
can)
sequence
a
can )ⁿ
has
THEOREM
Every
2
70
:
at
that
subsequences
THE
bounded
BOLZANO
-
to
converge
can )= ,
limits
different
diverges
,
then
,
can)
it
or
is
divergent
THEOREM
WEIERSTRASS
sequence
that
subsequence
one
has
a
convergent
subsequence
PROOF :
•
Io
•
•
can)
Since
let
with
the
Use
is
an C- Io
length
the
of
midpoint
bounded
for all
,
we
n
to
split
a
closed
bounded
and
C- IN
[l>
bee
Io
find
can
Io
o
into
]
2
,
each
with
length
'
til
.
interval
.
Atleast
•
Use
◦
midpoint
length
of
Atleast
◦
these
lot
(E)
" "
l
Io
will
this
construct
each
containing
(E) " l , with
into
of
closed
2
the
,
Can )
sequence
intervals
bounded
.
let
,
terms
many
Ik
interval
intervals
can]
we
infinitely
contain
the
these
of
1
Thus
split
to
sequence
•
will
Ikti
be
a
many
≥ Ii 7- Iz
.
.
.
of
terms
many
the
.
of
sequence
infinitely
infinitely
contain
closed
terms
The
the
of
Nested
bounded
and
and
sequence
Interval
Property
IIKI
intervals
length
each with
says
.
✗
NIK
-1-0
k=0
Choose
ÑIK
C-
a
✗
NIK
•
•
•
•
•
•
◦
contains
for
Then
This
We
N2
Using
induction
It
•
Prove
•
Given
him
anic
E. so
the
subsequence
PROPOSITION
Let
can )?=
( Azk )
-
lame
i.
i.
N
Anz
,
9=b
that
so
.
.
hkt
Ikx
:
to
converges
>
,
a.
, we
and
nk
infinitely
contains
,
Ania
-
sit
and
at
-
IN
C-
Itc
C-
✗
Ix
•
,
l
Choose
hi
s
.
-1
Ani C- I ,
.
still
have
Ankh
many
terms
C-
terms
in
1kt '
of
Ikti
an
to
after
so
from
choose
.
a
find
,
For KZN.it
•
=
this
an
ÑIK
C-
b
Suppose
OR
=o
nkt ,
do
excluded
have
bl
•
C- Iz
Anz
to
-
"
subsequence
a
.
(E)
≤
bl
la
choose
i
possible
is
we
sit
hi
-
number
one
it
choose
Choose
>
la
,
possible
to
want
le
all
only
is
exactly
It )Nl
.
C- Ilc
a
(
E
:
E
<
at
CE
came )✗
converges
to
a.
77
,
be
converge
a
to
sequence
L
,
can )
of
also
real
numbers
converges
to
It
-
L
.
subsequences
Caza 1)
-
✗
and
CAUCHY SEQUENCES
DEFINITION
A
78
sequence
lte
PROPOSITION
Every
> o
84
@ K)
Cauchy
both
are
> o
IR
< E
all
c
C- IR
so
,
Caktbk )
is
Cauchy
is
be
a
if
series
it
,
converges
to
series
{ 9k
a
real
number
CRITERION
of
sequence
numbers
real
.
The
is
convergent
:
) ( FN C- IN ) ( V-m.in
85
Cauchy
is
numbers
THE CAUCHY
:
only
COROLLARY
Clfeso)
for
ccak )
is
so
sequence
of
sequence
a
ant
83
COROLLARY
If
then
,
( bk )
and
THEOREM
(V-E
-
81
convergent
and
1am
N
≥
82
Every
it
Cauchy
is
LEMMA
let
mm
80
@ k )ⁿ
a
IN )
C-
bounded
is
sequence
PROPOSITION
If
:
( 7N C- IN ) ( V-m.in
)
Cauchy
Cclk )
If
it
79
PROPOSITION
If
Cauchy
is
C-
IN
)
m > n
≥ N
=) I anti tant
:
{ 9k
(3- NEIN )
is
convergent
,
then
:
•
IHNEN )
n ≥ N
/ Eakle
K=nt1
E
Z
tant }
t
.
-
am /
< E
g.
ABSOLUTE + CONDITIONAL
CONVERGENCE
THEOREM
If
80
:
£19k /
DEFINITION
[ an
87
88
Let
CAK)
If
lim
converges
,
:
absolutely
THEOREM
£9k
then
converges
ALTERNATING
THE
:
be
a
sequence
then
the
that
is
ak=0
£19k /
it
converges
Converges
SERIES TEST
of
real
{ C- 1)
series
Kak
A
90
convergent
Let
s
92
all
le
but
convergent
absolutely
not
convergent
conditionally
is
:
calc)
{ C- 1)
=
for
.
COROLLARY
suppose
91 '
:
£9k
series
≤
aktl
converges
k→x
DEFINITION
0 ≤
with
numbers
is
"
-
ale
Is
decreasing
a
for
and
-
Snl
≤
of
sequence
C- IN
n
iput
Sn
for all
an
real
=
numbers
É
to
converging
C- 1) Kak
then
0
.
:
C- IN
n
TESTS FOR ABS. CONVERGENCE
THE
RATIO
TEST
calc)
be
Let
①
If
②
③
If
L
If
1=1
LCI
>
the
,
1
a
the
,
,
then
sequence
series
series
the
of
real
{ 9k
£9k
Test
numbers
converges
diverges
is
such
that
absolutely
inconclusive
.
L=
¥3
%
'
exists
.
THE ROOT TEST
Let
calc)
① If
LL
② If
③ If
be
1
the
,
L > 1
L=I
,
Abs
L=¥fo
that
such
19kt " "
exists
.
diverges
series
Test
the
numbers
real
converges
series
the
,
of
sequence
a
inconclusive
is
REARRANGEMENTS
DEFINITION
We
→
that
say
there
it
99
exists
Inverse
bn
→
of
is
bijection
a
-
f
bijection
a
a
Cbn)
sequence
a
is
IN
:
IN
→
of
rearrangement
a
is
such
that
bn
=
the
Cak)
sequence
for
all
afcn)
n
C-
IN
.
bijection
a
rearrangement
of
rearrangement
of
iff
9k
9k
.
a
is
of
rearrangement
bn
.
.
DEFINITION
[ bk
100
is
calc)
a
being
{ bk
of
THEOREM
If
Proof
same
Let
f
The
partial
Sn
tin
IN
=
=
5
→
We
sum
=
Ebk
=
want
be
be
IN
EGK
the
of
imply
not
partial
bijection
a
sit
.
911-92
1-
bntbz
=
t
.
ak
.
.
_
tbn
Eak
noo
to
show
that
s=
him
n sx
tn
=
any
É
1C
an ,
.
.
then
,
of
rearrangement
a
:
Sn
lim
=
[also
.
,
sums
-
i.
does
absolutely
absolutely ]
Converges
Ébn
let
:
rearrangement
a
£9k
n=
Let
Cbk)
if
9k
is
rearrangement
a
of
sums
that
Ot
the
£9k
partial
sums
101
the
:
of Cala)
rearrangement
a
are
series
a
to
[
series
.
( ble)
→
the
bfckl
=
rearrangement
ak
/
,
K C- IN
of
£9k
Converges
Given
E >
since
=
there
,
Eak
> There
Put
An
o
Nz
is
It
tm
sum
The
then
N
≥
m
ibffs )
bfcz )
i
remaining
terms
É
i
51
-
19kt
have
that
±
≤
iflz )
11 )
{ 1-
all
for
E
Cauchy
.
,
.
.
,
[
≥ Nz
n
Ni
Criterion
Corollary
84 ]
f- CN }
bfcn )
_
Ot
term
Sn
É=,ak
=
is
included
the
in
partial
.
N
{
of
Itm
:
≥
n
,
the
satisfies
bk
included
are
Ent
in
kimtl
✗
We
EE
<
k=nt1
max
=
-
.
.
every
É=,b'k
=
M
,
Isn
sit
s.IE 19kt
IN
{ Mine }
Max
b 4- a)
C-
IN
C-
,
absolutely
converges
N=
=
N
is
Snl
-
/ akl
E
≤
Ice
≤
'
,
9k
.
ke
Ntl
SN
Itm
Ébk=
=
teÑÑ
[
an
]
1-
[
k=l
Itm
i.
for
-
SI
all
This
Itm
≤
M
m ≥
Snl
-
-
/
s
ÉETÉE
≤
=
E
.
the
that
means
Isn
1-
terms ]
remaining
limitm
s
=
MSN
THEOREM
102
£9k
If
of
£9k
↳ Separate
a
↳ Take
↳ Use
:
THE
is
whose
the
conditionally
tve
terms
convergent
x
terms
until
to
negative
a
terms
these
,
then
for
every
x
EIR
there
is
a
rearrangement
.
into
series
of
ve
is
sum
combination
-
REARRANGEMENT THEOREM
RIEMANN
will
we
reach
sum
exceed
terms
to
×
×
approx .sc
.
,
component
JCTIR
.
and
a
positive
terms
component
,
TOPOLOGY OF |R
OPEN
1-
SETS
CLOSED
DEFINITION
104
A
:
non
Ctx
G) ( Fr
c-
G
subset
empty
-
> 0
IR
Ot
C- IR)
) City
ly
is
-
it
open
al
:
YEG
er
Csc
An interval
(x
G
it
is
open
Example
(
:
9. b)
i.
Cx
-
r
empty
The
PROPOSITION
Prout
> Ctr
,
:
✗
K
set
108
If
U
≠ a
{x
-
r
PROPOSITION
xtr
,
109
:
nfi
it
Let
Gi
Put
i.
r
=
{x
i.
-
I
:/
done
÷.UGi
of
{r
,
≤
'
'
,
are
done
Then
×
,rz
R tr
}
Abi
,
.
sit
.
.
,
E
a
So
CK
}
> 0
x
-
empty
-
ri
,
-
Gi
,
definition
is
IR
open
of
is
not
true
.
.
.
d-
collection
of
i
ri
✗
+
,
112,3
>ctri
ri
}
there
open
open subsets
≤
Abi
,
.
.
)
is
-1-0
in
E
Gi
of IR
sets
open
that
=
is
is
.
collection
for
S.t.
{
C- Gi
assume
C- Gi
rn
sets
open
I
.
finite
be
.
the
subsets
open
of
non
is
any
On }
-
Friso
:
r
G.
:
> Fi C- I
Go
of
sit
I
no
collection
a
a re
is
collection
any
is
=
021
,
there
'
:
}
we
E
A Gi
min
on
.
> 0
of
iue
∅
open
is
}
intersection
=
C-
x
C.
)
{ Gi
let
:
X
b
U Gi
The
lying
neighbourhood
a
of
I
Union
E
G
open
open
:-c
x
) C-
setr
,
{ Gi
Take
,
Ca b)
let
Gi
ar
x
is
The
:
b-
,
E
)
[ E- I
Proof
a
-
has
r
neighbourhood
a
)
&
{
min
'' t r
called
is
,
>CEO ,
,
b
'
19
r
> 0
r
i
point
r
-
a
r=
]
every
ex
Put
rixtr
-
-
Gi
for all
i
open
.
r
> o
:
Remark
Open
The
:
sets
statement
Example
C-
A
:
110
PROPOSITION
let
:
111
If
:
But
112
let CIK )
)
[
=
the
is
not
Oil ]
non
in
F
-
r
-
open
10,1 ]
Proof
is
1lb
If
:
9 ≤
)
to
Cauchy
Cx
IN
F
-
K ≥N
IR
limit
seq
in
.
≠ Q
that
let
K C-
that
C- A
because
2)
There
y
A
in
is
€
Take
open
7x
XK
IN
=)
=
ik
G
,
<
for
r
all
KZN
.
it
Cauchy
every
sequence
Xk
l.mx/c
≤
a
≤
t.mx/cECasbJ
b
KSM
.
it and
open
RIA
,
C-
B.
=
not
C- A
ik
To
.
=D
only
its
if
IRIA
complement
is
closed
.
closed
is
IRIA
open
c-
gtthis
.
IN
sequence
B
that
x
is
not
can
find
a
closed
and
X
this
is
set A , but
:B
we
is
IRIA
=
in
in
shows
To show
ttrso
sit
B.
show
eventually
is
A.
r
is
Cauchy
a
then
is
set
F.
Let oc=hm
closed
IR
open
,
A
is
is
be
XK
Suppose
of
is
open
an
=/ ∅
A
CXK )
✗
A
,
is
G
act
-
closed
is
that
cab]
nor
open
A
there
it
Ktx
for all
subset
A
in
.
that
C-
the
.
for all
Ot
K ≥N
limit
a
risctr )
sit
C- IN
≤ 6
a
to
eventually
closed
is
Xk ≤ b
A
Suppose
If
many
G.
in
Sit
,
subset
b
neither
:
A
Assume
1)
infinitely
.
is
,
KS
THEOREM
d-
intersection
all
for
A
C-
Xk
converges
> 0
✗ fr
-
converges
,
that
limit
the
empty
a
≤
a
Fr
Cock )=
sequence
a
eventually
is
ca
that
Cock )= ,
of
E
be
We have
Ic
such
C- IN
open
is
[9lb ]
:
/
say
Cock )F= ,
A
:
We
sequence
IK
then
Example
'
It
,
N
definition
DEFINITION
1k
.
an
a
G
Since
the
By
'
≤ IR
A
the
:
for
k=l
exists
Proof
general
in
.
X
DEFINITION
hold
not
does
C-
B.
B
?
=
him
2k
impossible
is
closed
.
closed
point
y
with
ly
-
✗
Icr
and
For
each
we
B
not
is
COROLLARY
117 !
The
COROLLARY
118 :
The
Both
CUAI )
IR )
( nai )
DEFINITION
of
d-
Union
lyle
with
yk
yle
sequence
finite
any
-216
'
1K
with
13
in
closed
collection
closed
of
,
9k€
hmyk
YKEB
A
€1B
x
=
is
closed
sets
is
closed
.
of
statements
open
sets
,
-1hm
116
:
'
CIRIAI )
A
sets
corresponding
from
Laws
Ot
collection
any
follow
=
UCIRIAI )
=
call
we
:
119
a
intersection
Morgan's
IRI
is
closed
statements
De
and
there
,
constructed
have
i.
IN
C-
K
D-
Closed
=
subsets
n
the
5-
that
IR
in
closure
of
contain
A.
set A.
This
is
the
at all
intersection
•
F
is
A
closed
C-
F
A C- Ait
is
the
If
B
is
Example
Ñ
and
:
closed
smallest
closed
a
Ai
=
closed
is
that
set
Coil )
Ñ
,
that
set
=
A
contains
A
contains
.
A- ≤ B.
linen
A- 3
[ 0,1 ]
AJ
Az
PROPOSITION
120
{
=
A- 2
'
A
:
2) If
}
A
A
A
other
=
=
closed
is
that
the
.
IR
E
Proof :D Assume
=
K C- IN
,
IN
=
[ ON Az]
=
On
/K
IN
=
hand
is
,
itt
closed
we
.
A
.
Then
always
=
A
A-
a
is
closed
set
AEÑ
have
containing
Ñ
Ñ
,
therefore
D-
is
trivially
closed
,
because
A-
is
closed
.
A
→
A-
≤
A
LIMIT
POINTS
DEFINITION
1) We
121
that
say
sequence
2) The
A
It
•
✗
A
is
=0
✗
→
123
C- A
let
→
.
Let
•
•
•
'
For
We
that
•
For
•
i.
is
there
a
of A
set
derived
the
.
each
j
-
IR
x
EN
if
:
, we
limit
of
2=0
of A-
point
limit
9
limit
not
are
not
is
point d-
of
points
A
,
.
then
called
is
x
'
A
calc)=
sequence
a
in
'
AVA
and
A
AVA
=
sequence
sequence
x
can
EA
find
,
we
F
so
,
in
,
A
C- F.
I
≤ A-
AUA '
in
Nj
are
.
This
will
sequence
to
converge
some
limit
x
done
C- IN
strictly increasing
,
either
ixnj
-
< '
xnj /
lj
2k
then
,
of A
containing
e- F
to
converges
of A
A-
,
Cauchy
a
'
XK
sit
-
lock -21
( )
nj
sequence
-8=1
<
'
Ot
for all
12J
natural
K
numbers
≥
Nj
sit
.
.
Ian's
-2C
<
.
I Clnj
x1
-
a
cases
that
Iceni
C- IN
the
Cauchy
a
A
choose
j
of
to
¢2M )
X
isolated
closed set
is
it
'
.
C- IN
j
point
≤
:
A
C- AUA '
x
&
x
In both
such
:
A-
AVA
be
can
A
that
CXK)F= ,
for all
•
it
112
A
C-
x
point
a
every
ot
and
limit
a
point
point
C- A
is
limit
the
closed
each
is
C-
a
is
Assume
isolated
'
of
point
converges
limit
and
an
any
( 1k )
a
limit
✗
×
be
is
x
Show
→
called
is
limit
a
:
is
If
For
that
AUA
'
.
↳
F
]
a
x
follows
It
→
:
126 :
Suppose
Then
≤
.
'
( 1,2 )
C-
2C
THEOREM
→
A
set
a
co )
✗
point
I
,
n¥
is
DEFINITION
Co
€
✗
C-
in
limit
a
point
Proof :
x
A
of
points
not
is
x
eventually
For all
•
to
converges
limit
of
point
{ 23
U
then
,
ze=2
•
limit
a
that
ot all
'
]
I
,
be
Every
•
is
ALEX }
( 0
2C
Must
•
C- IR
:
Co
=
x
in
set
Example
:
,
s
xnj
the
is
'
12J
C- A
limit
or
of
a
✗
nj
C- A '
sequence
of
points
in
A
i.
we
can
find
anj.CA
'
tis
EIR
.
i.
•
Since
•
'
•
2
.
is
$
I
A-
the
limit
A
① If
A
② It
A
127
is
is
:
is
of
consists
Corollary
X
,
of
let
sequence
a
a
limit
isolated
A
be
bounded
above
bounded
below
a
then
,
,
then
in
A
point
of
points
of
non
-
.
A
.
.
A
:
x
and
subset of
empty
A-
sup IA )
C-
int (A)
C- A-
.
EA
limit
IR
'
points
of
A-
COMPACT
SETS
LIMITS AND CONTINUITY
LIMITS
DEFINITION
limit
FUNCTIONS
OF
132
let
:
of
point
A
f-
be
We
.
function
a
flx)
that
say
defined
on
a
non
l
to
converges
-
as
A ≤ IR
set
empty
✗
→
if
a
l
.
that
Suppose
is
a
is
number
real
a
el
satisfying
( te
LEMMA
135
)
> o
It
:
( 78
>
o
)
( Hoc
for
f- ( × )=c
A)
C-
O
NEA
all
Ix
C
-
at
11-1×1
flx )
Iim
then
,
8
<
✗
→
=
Ll
-
<
E
for
c
C- A
A
any
a
'
CEA
>
LEMMA
136
LEMMA
137
f- ( x )
If
:
=
suppose
:
for
≥
that
him
*
→a
SEE
It
①
②
L >
It
a
,
Leo
THEOREM
of
0
,
138
let
lm
l.im
xsacfcx )
eim
②
→
✗
a
A)
0
( 75>0 )
Ctx
c-
A)
O
fcx )
functions
be
g
✗→ a
THEOREM
IHEOREM
140
that
THE
:
TEST
SEQUENTIAL
let
It
f-
be
there
f-
a
exists
above
( x ) =L
to
converges
C
gcx
-
al
Ix al
8
e
-
as
,
8
C
then
=
non
a
fcx )
=)
-
fcx )
>
co
set
empty
exists
THEN=
lim
✗
→a
fcx)
DNE
has
that
a
limit
point
:
f-
eim
✗→
.
✗
Cx)
≤
flock )
for
gcx )
for
=L
all
oc
C- A
,
then
function
m
in
,
Alsea }
DIVERGENCE
defined
sequence
2
≤
CIK)F=
sequence
every
L
a.
FOR
a
iff
It
:
on
a
?,
Gaelic
set
nonempty
in
A) { a }
A
that
that
has
converges
limit
a
to
a
0€
there
'
> o
diverges
it
A
mʰ
the
Lima
-
A
c-
a
any
Lm
=
Given
:
a
✗ C-
for
9
=L + m
=
,
139
✗→
=
CL
=
¥¥
lim
④
Ix
)=M
✗→ a
fcxlglxl
xs a
C.
defined
lim
and
=L
fcx )
eim
=L
C-
f.
then
,
( tix
[ fcxltglx ) ]
l.im
③
fcx )
C- A
A
✗→a
①
✗
( 78>0 )
then
:
If
.
then
all
sequences
in
A'
{a }
that
have
different
limits
point
such
a.
that
flock) :-.
,
THE
THEOREM
SQUEEZE
Suppose
there
FUNCTIONS
FOR
exists
function
2
gcx]
Then
it
ein
gcx )
✗→ a
DEFINITION
144
Then
a.
let
:
him
1-1×1
→a
145
( 78
C- IR )
Cte >
_
o
hcx )
> o
We
=L
xs a
defined
on
a
to
mean
) lfxEA )
o
non
that
empty
-
A
set
that
has
limit point
a
:
fcx )
Isc ales
e
-
a
define
¥5b
A)
C-
on
to
=L
8
<
-
defined
fix )
( toe
be at
0C
function
be
CFM C- IR )
)
flat
1in
then
,
( 78>0 )lV-xEA )
f
above
≤
:
M
>
means
let
:
bounded
not
¥?
tlx)=
-
ltfm
DEFINITION
be
c- IR )
=
flx)
lim hcx )
=
(x )
f-
Ctm
that
Zs a
define
we
similarly
=L
≤
such
2m
that
L
is
lfcx)
m
-
LI
number
real
a
that
IR
A ≤
set
non-empty
a
mean
x ≥
flat
=)
is
satisfying
LE
CONTINUITY
DEFINITION
f-
146
let
:
continuous
is
f
f-
continuous
is
if
is
a
flat
↳
↳
f- (a)
empty
A
on
at
point
of
Ix
/
is
a
require
we
-
/
a
<
in
point
every
whenever
A
IR
≤
and
a
suppose
EA
We
.
that
say
:
at
a
A
set
I
8
A
,
then
A
ot
-
flat
that
say
we
point
isolated
an
11-1×1
/
<
f
is
E
0nA
continuous
.
:
defined
fcx )
✗ → a
it
a
-
( 78>0 ) / txt A)
A
on
limit
a
is
em
↳
at
continuous
is
non
a
A
on
t.VE > o )
tf
be
f-
must
lim
=
exist
f- Cx )
Sega
-
IHEOREM
a
U
limit
{b }
147
:
point
≤
let
of
be
f
and
A
B.
a
It
function
that
function
a
b.
defined
%?
=
is
g
dim
xs a
%
lim
✗→ a
gltcx) )
=
gcb)
=
on
fcx )
non
a
exists
continuous
)
9 [ fcx ]
g (
him
Isa
=
fcx)
.
B
B
at
gcb)
A ≤
set
empty
let
on
)
-
be
b
,
any
then
:
IR
set
•
Suppose
satisfying
that
{
a
fcx )
C:
IR
x
is
c.
A)
.
COROLLARY
THE
148
The
:
SEQUENTIAL
let
CRITERION
be
g
Then
defined
continuous
COROLLARY
Cock)iP=
The
:
function
B
in
,
non
empty
-
C-
b
in
151
let
:
f
A
:
dim
that
continuous
xk
5While
the
pre
-
→
of
image
→
f-
←
(c)
C-
B
(D)
≤
A
THEOREM
THE
If
154
EXTREME
b
=
f-
f-
be
number
y
between
UNIFORM
DEFINITION
continuous
=
y
g
159
on
is
CD)
f-
C- C
B
≤
is
of
sequence
a
to
converge
set
a
c
≤
glb)
A
:
flx ) C-
:
on
}
D
set
compact
a
K
→
f-
then
,
(K)
compact
is
on
a
compact
set
closed
and
1C
,
f
then
its
attains
maximum
and
:
function
on
and
f- Ca )
f-
a
Cb )
,
there
bounded
atleast
is
interval
number
one
For
[a ,b]
C
.
C-
each
Carb]
.
A
not
let
it
f-
be
a
function
defined
set
a
on
AEIR
-
We
that
say
f
is
:
) ( 78>0 )
( ta
uniformly
continuous
> o
CFE > o)
( to
is
}
continuous
is
f-
not
:
{ XEA
=
exists
does
,
under
image
there
it
b
Cock ) ,?
b-
CONTINUITY
( V-E
f
D
✗
:
at
B
on
The
.
to
.
continuous
a
f- Cc)
with
tf
K
on
.
:
INTERMEDIATE VALUE THEOREM
let
←
{ fix)
=
continuous
is
values
minimum
THEOREM
IR
≤
converges
but
function
a
set
function
a
VALUE
function
a
THE
If
:
(c)
a
f-
g-
be
B
→
B
:
.
les
DEFINITION
continuous
set
itf
B
B
not
is
g
that
such
is
glad =gCb )
Cock)= ,
sequence
150
a
at
B
him
every
on
on
KS
for
function
continuous
CONTINUITY
FOR
function
a
is
g
of
composition
> o
A)
C-
) ( 7. a
c-
Ix al
-
on
A) (Fx
a
C-
<
set
A)
lfcxl
8
A
,
then
f- (a) I
<
E
:
Ix al co
-
-
and
lfcx)
-
flat /
≥ E
uniformly
.
f-
uniformly
not
is
( Fe
PROPOSITION
E >
160
and
0
IFCXK)
0
>
)
A
THEOREM
102
continuous
function
≥
E
If
:
1C
on
) ( Fak
IN
C-
pair
a
f- Cyk ) I
-
( UK
:
it
continuous
f
sequences
for
all
K
lock
A)
C-
CXKIP
C- IN
f
:
uniformly
not
of
it
only
A) [Jock
C-
is
function
a
and
9k
-
continuous
[
and
yet ?
/
it
on
A
iff
flak)l
-
"m
( KK
k→
-
≥
E
exists
there
.
S.t.
A
in
lflxk )
and
YK )
=
an
and
O
.
continuous
is
on
K
set
compact
a
then
,
f
uniformly
is
.
DERIVATIVES
DEFINITION
f-
103
let
:
be
f
differentiable
is
it
function
a
lim
f
differentiable
is
THEOREM
164
continuous
It
:
DEFINITION
165
f-
:
has
has
f
A
in
point
is
Ibb
maximum
FERMAT 's
f-
local
point
THEOREM
be
a
f
We
say
the
set
that
differentiable
is
A
on
IR
≤
at
on
point
a
9
A.
C- A. then
f
either
f
is
be
CEA
on
A
-
is
at
A
<
clc
8
if
A
C-
8
=
if
A
:
f- Cc)
≤
:
flx)
>
critical
or
C-
fcx )
=)
C
a
C
C
at
≥
off
point
dim
tcx )
✗ SC
X
f- (c)
-
-
1- (c)
on
0
=
it
A
.
C
:
function
or
maximum
of
f
then
on
at
A
on
)lV-xEA )
say
differentiable
not
let
> o
We
:
,
differentiable
is
local
a
minimum
178
DEFINITION
.
a.
local
a
C- IR
A
set
q
-
178>0 ) ( tx .CA/lx-c1
f
non-empty
a
flat
-
x
every
at
at
A
on
flx)
→a
function
a
on
:
✗
If
defined
on
A
a
.
defined
local
minimum
on
a
on
non-empty
A
at
a
set
point
A
≤
C
IR
C-
.
A
If
,
t
then
has
C
either
is
a
a
critical
Rollie's
let
It
THEOREM
be
ff
at
function
a
that
differentiable
is
least
point
one
the
on
C-
C
continuous
is
on
Caib)
interval
open
Caib )
compact
a
with
g-
fca)
and
(c)
'
carb]
interval
f- Cb )
=
.
then
,
there
exists
0
=
^
y
:
I
1
I
1
I
1
'
I
a
MEAN
/ HE
let
It
f
a
function
differentiable
[ a . b)
with
that
is
the
on
b
open
continuous
interval
'
(c)
f- (b)
=
-
b-
the
this
,
of
graph
Cb f- Cb) )
,
on
( a. b)
there
exists
at
least
point
one
theorem
says
y=fCx)
that
fla )
there
cccifcc ) )
at
is
is
parallel
T
•
•
A
:
then
.
a
:
Equivalently
,
Cab]
interval
compact
a
:
f-
Geometrically
x
THEOREM
be
is
C-
C
C
VALUE
f
f
e
fcb)
-
tea ]
'
=
f' (c)
Cb
-
a)
B
C
C-
( aib )
to
the
sit
.
secant
the
line
line
tange
through
to
Caifca )
)
and
CAUCHY 'S
let
If
MEAN
f-
and
point
g
Carb )
( fcb)
L' HOSPITAL 's
-
that
continuous
are
differentiable
are
C-
C
THEOREM
functions
be
g
t and
both
one
VALUE
with
fla )
)
/
g
the
on
on
interval
open
( a. b)
Caib ]
interval
compact
a
.
exists
there
linen
least
at
:
(c)
( gcb)
file )
=
gla) )
-
RULE
suppose
If
¥5c
1- (c)
glc)=
=
,
for
0
exists
[
CE
then
a.
f- 1×1=10
'
but
b)
.
:
l.im
=
f (c)
✗→ C
1- (c)
✗→ c
g' cc )
him
9k)
'
SEQUENCES OF FUNCTIONS
POINTWISE
DEFINITION
A
≤
IR
US
172
We
.
(
Let
:
fk )
be
i
lfk )P=
that
say
CONVERGENCE
UNIFORM
.
converges
,
lim
fuk)
ks
We
DEFINITION
170
set
THE
A
A
>
a) LIN
178
≤
IR
CAUCHY
sequence
.
( fk )
let
:
If
C- IN
fk
each
of
functions
) (FN
C- IN
a
a
sequence
of
functions
f
A
function
a
on
non-empty
a
it
A
A
on
then
,
it
AEIR
on
) ( V-m.rs
uniformly
c-
IN )
it
( b- ✗
C-
A)
We
say
-
fix ) /
CE
.
so
and
.
:
Ifklx)
≥N
functions
of
defined
that
is
converges
the
uniformly
limit
f.
=
lfmlx )
to
a
CONVERGENCE
converges
set
:
A.
on
,
on
k
convergence
sequence
continuous
is
lfklic?
of
A) IV-kc.IN )
c-
on
f
function
a
limit
to
pointwise
UNIFORM
FOR
> o
( tx
be
i
CRITERION
( te
)
defined
flx)
=
uniformly
to
pointwise
pointwise
be
implies
convergence
-
IHEOREM
(fkIÑ=i
converges
,
life
Uniform
the
is
let
:
Cftc )F=
that
fcx )
that
say
functions
of
sequence
a
only
it
min ≥
:
N
>
-
fnlxl /
LE
function
f
on
a
THE
DIFFERENTIABLE
( f- KIK? ,
The
→ IR
1- Cx)
'
limit
pointwise
[ aib]
fk)
(
of
✗
derivatives
glx)
=
interval
compact
on
the
is
of
sequence
Then
THEOREM
differentiable
is
:[ a . b)
f
LIMIT
If 'kIk=
for
all
i
uniformly
converges
,
.
to
g
:
[
]
a. b
→ IR
.
C- Carb ]
I
SERIES OF FUNCTIONS
DEFINITION
184
We
:
§;fk
that
say
absolutely
converges
A
on
§
it
lfkcx) /
for
converges
,
all
C-
2
A.
A
DEFINITION
185
THE
WEIERSTRASS
suppose
all
convergent
§,fk on A-
to
it
its
fklxl
≤
A
on
M TEST
-
exists
of
sequence
a
(Mk )k%
numbers
with
✗
If the
partial
of
sequence
for
Mk
KEIN
all
and
A.
E
x
uniformly
is
uniformly
converges
,
there
that
§ ,fk
series
Ifk ) ?
(
sums
The
:
uniformly
numbers
of
series
on
Mk
converges
,
,
then
the
functions
of
series
§
,
fk
absolutely
converges
and
A.
POWER SERIES
DEFINITION
190
let
:
IR
E
Xo
and
[ 9k )ii=o
let
be
a
of
sequence
real numbers
.
We
call
✗
i. =
a
the
DEFINITION
case
191
:
xo=o
The
{
THEOREM
→
number
→
the
If
the
"
Go
=
1- 91
coefficients
%
:
KEIR
:
(x
-
Xo )
Azcx
t
90,91192193
akcx)k=
Aix
Got
.
.
-
)Z
t
Xo
.
.
.
.
92×2
t
t
93×3 t
.
.
.
,
of
domain
of
convergence
[
1<=1
that
r
a
power
satisfying
interval
power
t.sc x.co/7lXz-xol
-
Xo )
akcx xD "
-
a
power
converges
is
series
:
}
195
Suppose
on
-
with
series
power
In
AKCX
,
[ Xo
-
series
series
0
C rc
when
converges
IX. Kol
-
,
the
x=xc
where
≠ Xo
converges
series
power
Xi
.
Then
for
absolutely
any
and
real
uniformly
riko tr ]
diverges
when
x=x z
,
then
it
also
diverges
for
all
×
satisfying
DEFINITION
196
The
:
radius
R
the
It
①
197
him
It
9kt
1cg
②
If
/
k,
A
-
-
for all
be
"
xo)
exists
'
/
""
exists
Series
defined
is
(a)
follows
as
all
C- [
x
xo=o
,
Xo
r
-
find
we
TAYLOR'S THEOREM
f
each
:[ Xo
x
-
r
,
C-
rxotr
,
the
xotr ]
→
Exo rixotr
-
]
""
radius
times
differentiable
R
convergence
{
1- '
¥)
FORM
Series
gives
us
ojk
:
REMAINDER
THE
function
a
there
,
✗
xk
OF
be
IR
-
]
"
1C =L
(x
!
exists
=
that
a
point
fcnt
"
is
nti
between
C
cc)
↳c- Xo)
(ntl ) !
×
and
on
No
the
interval
such
that
:
" '
:
f-
(x )
=
[
flkttxo )
(x xojk
-
k!
1+
"" "
(c)
Intl )
(x
-
xojnt
'
!
k=O
TAYLOR 'S
If there
.
"K
9k
( Xo)
K
,
Rnlx)
This
of
with
:
Maclaurin
LAGRANGE 'S
WITH
f-
=
%=
fix)=
For
R=
:
9kt '
him
x
let
}
Ak
ksro
f-
When
×
'
lim
R=
,
.
series
power
R=
,
IR
at
:
SERIES
Taylor
for
a
is
series
converges
E
x
ks
an
"
1C =L
a,
lim
TAYLOR
É,akCx
power
a
:[ akcx.ro)
Ix xol
converges
let
:
of
convergence
{
sup
=
series
power
PROPOSITION
of
INEQUALITY
exists
an
mso
If
5.1
.
/ Rnlx )
≤
" ">
Call
Mrn "
Cnt 1) !
≤ m
for
all
x
C-
[ xo - r. xc . i r ]
,
then
:
[ Xo
-
rixotr ]
.