Download Introduction to Real Analysis

Introduction to Real
Analysis
Dr. Weihu Hong
Clayton State University
9/17/2009
Theorem 2.2.6






1
lim p  0
n  n
(a) If p>0, then
1
n
n p  lim p  1
(b) If p>0, then lim
n 
n 
1
n
n 1
(c) n n  lim
n 
n
0
n
(d) If p>1 and α is real, then lim
n  p
(e) If |p| < 1, then lim  p n  0
lim
n
n
pn
0
(f) For all pєR, lim
n  n!
Monotone Sequences


{an }n 1 of
Definition 2.3.1 A sequence
real numbers is
said to be
(a) monotone increasing (or non-decreasing) if
an  an1 for all n  N ;

(b) monotone decreasing (or non-increasing) if
an  an1 for all n  N ;

(c) monotone if it is either monotone increasing or
monotone decreasing
Examples


{
a
}
1. Define the sequence n n 1 as follows:
a1  2，and an1  2  an

Is the sequence monotone increasing or monotone
decreasing?
2. Define the sequence {an }n 1 as follows:
an  p n , where p  (0,1)

Is the sequence monotone increasing or monotone
decreasing?

n
{
a
}
3. Define the sequence n n 1 as follows: an  2  (1)
Is the sequence monotone increasing or monotone
decreasing?
Theorem 2.3.2


{an }n 1
If
is monotone and bounded, then it
converges.

{
I
}
Corollary 2.3.3 If n n1 is a sequence of closed
and bounded intervals with I n  I n1 for all nєN,

then
I
n

n 1

Note: The intervals must be closed in Corollary
2.3.3
Infinite Limits

{an }n 1
Definition 2.3.6 Let
be a sequence of real
numbers. We say that {an }n 1 approaches infinity,
or that {an }n 1 diverges to ∞, denoted
an   or lim an  
n 
if for every positive real number M, there exists an
integer KєN such that an  M for all n  K

How would you define a sequence approaches to
−∞?
Theorem 2.3.7
{an }n 1

If
is monotone increasing and not bounded
above, then an   as n  .

Proof: Since the sequence is not bounded above,
therefore, for every positive number M, there exists
a term aK such that aK  M . Since the sequence
is increasing, thus,
an  aK  M for all n  K
Therefore,
an   as n  .