Introduction to Real Analysis

Dr. Weihu Hong

Clayton State University

9/18/2008

Theorem 2.3.2

If is monotone and bounded, then it converges.

Corollary 2.3.3 If is a sequence of closed and bounded intervals with for all nєN, then

Note: The intervals must be closed in Corollary 2.3.3

1}{ nnI

1}{ nna

1 nn II

1nnI

Infinite Limits

Definition 2.3.6 Let be a sequence of real numbers. We say that approaches infinity, or that diverges to ∞, denoted

if for every positive real number M, there exists an integer KєN such that

How would you define a sequence approaches to

−∞?

1}{ nna

1}{ nna

1}{ nna

n

nn aora lim

KnallforMan

Theorem 2.3.7

If is monotone increasing and not bounded above, then

Proof: Since the sequence is not bounded above, therefore, for every positive number M, there exists a term such that Since the sequence is increasing, thus,

Therefore,

1}{ nna

. nasan

.MaK Ka

KnallforMaa Kn

. nasan

Subsequence of a sequence

Definition 2.4.1 Given a sequence in R,

consider a sequence of positive integers

such that . Then the

sequence is called a subsequence of the

sequence .

}{ np

}{ kn

}{ np

}{kn

p

321 nnn

Examples of subsequences of a sequence

Consider a sequence . Let be

two sequences of positive integers. Then we have

two subsequences and of the sequence,

one of which is consisting of all the terms from the

sequence with odd indices while the other one

is consisting of all the terms from the sequence

with even indices.

}{ np

11 }2{,}12{ kk kk

}{ np

}{ np

}{ 12 kp }{ 2kp

Subsequential limit of a sequence

Given a sequence .Let a be either a real

number or ±∞. We say that a is a subsequential

limit of the sequence if there exists a

subsequence such that

}{ np

}{kn

p

kasapkn

}{ np

Example of subsequential limit

Consider the sequence . Is a = 2 a

subsequential limit of the sequence? Is a = 0 a

subsequential limit of the sequence?

Consider the sequence . Is a = +∞ a

subsequential limit of the sequence? Is a = -∞ a

subsequential limit of the sequence?

1})1(1{ n

n

1})1{( n

nn

Theorem 2.4.3

Let be a sequence in R. If converges

to p, then every subsequence of

also converges to p.

}{ np }{ np

}{kn

p }{ np