Download Introduction to Real Analysis

Introduction to Real
Analysis
Dr. Weihu Hong
Clayton State University
10/6/2009
Contractive sequences

{
p
}
n n 1 in R is
 Definition 2.6.6 A sequence
contractive if there exists a real number b, with 0 < b
< 1 such that
| pn1  pn | b | pn  pn1 |
for all nєN with n≥2.
Theorem 2.6.7

Every contractive sequence in R converges in R.

Furthermore, if the sequence { pn }n 1 is contractive
pn  p， then
and lim
n 
b n 1
(a) | p  pn |
| p2  p1 |, and
1 b
b
(b) | p  pn |
| pn  pn 1 | .
1 b
where 0  b  1is the const in Definition 2.6.6.
Series of Real Numbers

{ pn }n 1 be
Definition 2.7.1 Let
a sequence in R, and

{ pn }n 1 ,
let {sn }n1 be the sequence
obtained
from
n
where for each nєN, sn   pk .The sequence {sn }n1
k 1
is called an infinite series,
or series, and is

pk or p1  p2    pn  .
denoted either as 
k 1
For every nєN, sn is called the nth partial sum of
the series and pn is called the nth term of the
series.

p
k 1
k
converges / diverges  {sn }n 1 converges / diverges

s   pk  s  lim s n
k 1
n 
Examples


k
r
(a) Geometric series 
k 1



1
(b) Consider the series 
k 1 k ( k  1)
(c) Consider the series

1

p
k 1 k
.
Theorem 2.7.3 (Cauchy
Criterion)


p
The series
k 1
k
converges if and only if given ε>0,
there exists a positive integer K such that
m
p
k  n 1
k
  for all m  n  K
Corollary 2.7.5


If
p
k 1

k
Remark.
If lim pk  0
k 
converges, then
lim pk  0
k 
Is the following statement true?
, then

p
k 1
k
converges.
Theorem 2.7.6

Suppose pn  0 for all nєN. Then

p
k 1

k
converges  {sn } is bounded above.
Why the above theorem doesn’t apply to the series

k 1
(

1
)

k 1
Structure of Point Sets



Definition 3.1.1 Let E be a subset of R. A point
pєE is called an interior point of E if there exists an
ε>0 such that N ( p)  E
The set of interior points of E is denoted by Int(E).
Definition 3.1.3


(a) A subset O of R is open if every point of O is an
interior point of O.
c
F
 R \ F is open.
(b) A subset F of R is closed if