1
Dr. Bradley
Quoc Phung
230A
Theorem Limit of Sequences
Theorem:
Suppose sn  and

lim k .sn  k .s

lim( sn  tn )  s  t

lim( sn .tn )  s.t

lim
tn  are real sequences with
lim sn  s , lim tn  t
x 
x 
x 
x 
x 
x 
1 1

sn s
Sn  0 and
Provide
S  0
Prove: lim( sn .tn )  s.t
x 


Let  > 0.
sntn  st  sntn  snt  snt  st
 sn tn  t  t sn  s

Since lim tn  t ,
x 
 N1 such that n > N1 
tn  t 

2M
when | Sn | < M for all n.

Since lim sn  s ,
x 
So.
 N2 such that n > N1 
sntn  st  sn tn  t  t sn  s
M

Therefore:
sn  s 

2

2M

lim( sn .tn )  s.t
x 

2
t

2  t  1

Q.E.D.

2  t  1
2
Dr. Bradley
Quoc Phung
230A
Theorem 1:
If sequence { an }is convergent , then { an } is bounded.
Solution:
*
We need to prove
*
| a n| = | a n

Assume { an }is convergent to a
  0,
| a n| < M ,
for all n  N.
- a+ a|  |an -a|+|a|

lim an  a
n 
 N* such that
Let  = 1, then  N* such that

Therefore,
| a n|
n > N* 
| an - a | < 1
|an

|an -a|+|a|
1
Let M = Max { a1 , a2 , … , a n-1 ,
Then | a n | 
Therefore:
| an - a | < 
=
=

n > N* 
- a+ a|
+
|a|
1+|a|}
M
for n > N*
If sequence { an }is convergent , then { an } is bounded. Q.E.D.
3
Dr. Bradley
Quoc Phung
230A
Theorem 2:
If a sequence { an }is bounded, monotonic , then { an } is convergent.
Solution:
* We need to prove:
lim an  L
n 


  0,
 N* such that
n > N* 
| an - L | < 
Suppose a sequence { an }is bounded and monotone increasing.
Let A = { an | n  N } be a nonempty bounded set.
Then by the Completeness axiom:
If every nonempty subset A of R is bounded above,
then subset A has a least upper bound.
Therefore
A = { an | n  N } has a least upper bound  Sup A = L.

Sup A -  = L -
Sup A = L
aM
Since L= Sup A, then L-  is not Sup A.
So there exist M such that L -   aM  L

A sequence { an }is monotone increasing 

From ( 1 ) and ( 2 )
i e:
  0,
that L -   a n
 M such that

L -   aM
aM  a n  L
(1)
for n > M.
aM  a n  L  L +  ( 2)
 L+

| an - L | < 
n>M

| an - L | < 
Therefore: If a sequence { an }is bounded, monotonic , then { an } is convergent Q.E.D.
4
Dr. Bradley
Quoc Phung
230A