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Introduction to Real
Analysis
Dr. Weihu Hong
Clayton State University
9/18/2008
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  .
Subsequence of a sequence

Definition 2.4.1 Given a sequence
consider a sequence
such that
{nk }
sequence { pn } .
in R,
of positive integers
n1  n2  n3  
sequence { p }
nk
{ pn }
. Then the
is called a subsequence of the
Examples of subsequences of
a sequence

Consider a sequence { pn } . Let {2k  1}k 1 ,{2k}k 1 be
two sequences of positive integers. Then we have
two subsequences { p2 k 1} and { p2k } of the sequence,
one of which is consisting of all the terms from the
sequence { pn } with odd indices while the other one
is consisting of all the terms from the sequence { pn }
with even indices.
Subsequential limit of a
sequence

Given a sequence { pn } .Let a be either a real
number or ±∞. We say that a is a subsequential
limit of the sequence { pn } if there exists a
subsequence { pnk }
such that
pnk  a as k  
Example of subsequential limit

Consider the sequence {1  (1) n }n 1 . Is a = 2 a
subsequential limit of the sequence? Is a = 0 a
subsequential limit of the sequence?

n

{(

1
)
n
}
Consider the sequence
n 1 . Is a = +∞ a
subsequential limit of the sequence? Is a = -∞ a
subsequential limit of the sequence?
Theorem 2.4.3

Let { pn } be a sequence in R. If
{ pn } converges
to p, then every subsequence { pn } of
k
also converges to p.
{ pn }