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Introduction to Real
Analysis
Dr. Weihu Hong
Clayton State University
10/2/2008
Cauchy Sequences


{ pn }n 1 in
Definition 2.6.1 A sequence
R is a Cauchy
sequence if for every ε>0,there exists a positive
integer no , such that pn  pm   for all integers n,
m ≥ no .
Remark.
pn  pm   , n, m  N and n, m  no

pn  k  pn   , n  N and n  no , k  N
Theorem 2.6.2


(a) Every convergent sequence in R is a Cauchy
sequence
(b) Every Cauchy sequence is bounded.
Theorem 2.6.3

If
{ pn }n 1
is a Cauchy sequence in R that has a
convergent subsequence, then the sequence
converges.
{ pn }n 1
Theorem 2.6.4

Every Cauchy sequence of real numbers
converges.

Remark. This theorem actually states that R is
complete.
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.