EE603 Class Notes

10/24/2012

John Stensby

Appendix 11A: Limit, Limit Superior, and Limit Inferior of a Real Number Sequence A sequence of real numbers is a mapping from the integers into the real numbers. R denotes the set of real numbers; x ∈ R ⇒ -∞ < x < ∞. The set of extended real numbers is denoted as R +, and it is R∪{±∞}. We denote a sequence by the notation X(n). Often, the notation Xn is used; in the literature, other notational conventions have been used to represent sequences. Examples: 1. X(n) = 1/n, n ≥ 1. 2. X(n) = sin(nπ/10) 3. X(n) = n2. Basic Concepts A sequence may have a limit. X0 ∈ R is a limit of sequence X(n) if, for each ε > 0, there exists an integer Nε such that ⎮X(n) - X0⎮ < ε for all n > Nε. Note that Nε depends on ε. If one exist, a limit is unique. A sequence may diverge to either +∞ or -∞. Example 1 above has the limit X0 = 0. Examples 2 does not have a limit; it oscillates forever. Finally, Example 3 diverges to +∞. A monotone increasing sequence is a sequence for which X(n+1) ≥ X(n). A bounded sequence has the property that ⎮X(n)⎮ < M for all n, where M is some positive, finite number. It is obvious that a bounded, monotone increasing sequence X(n) is convergent to some real number X0. We write X(n) ↑ X0. Also, for a monotone increasing sequence, we write X 0 = limit X(n) = Least Upper Bound {X(n), − ∞ < n < ∞} n →∞

= supremum {X(n), − ∞ < n < ∞}

(11A1)

= sup{X(n), − ∞ < n < ∞} .

The terms Least Upper Bound, supremum, and sup mean the same thing.

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11A-1

EE603 Class Notes

10/24/2012

John Stensby

A monotone decreasing sequence is a sequence for which X(n+1) ≤ X(n). It is obvious that a bounded, monotone decreasing sequence X(n) is convergent to some real number X0. We write X(n) ↓ X0. Also, for a monotone decreasing sequence, we write X 0 = limit X(n) = Greatest Lower Bound {X(n), − ∞ < n < ∞} n →∞

= infinium {X(n), − ∞ < n < ∞}

(11A2)

= inf{X(n), − ∞ < n < ∞}

The terms Greatest Lower Bound, infinium, and inf mean the same thing. Cauchy Sequences ⇔ Convergent Sequences A sequence of real numbers is said to be Cauchy if limit X(n) − X ( m ) = 0 .

(11A3)

n, m →∞

In (11A3), integers n and m approach infinity; the manner in which they do this is not important. For a Cauchy sequence, the terms get "closer together" the "farther out" you go in the sequence. A major theorem in the theory of real analysis is that a real sequence is convergent if, and only if, it is Cauchy. Because of this, we say that the real numbers are complete (or the real numbers form a complete vector space). To test a sequence for convergence, it is not necessary to "cook up" a limit. All we have to do is show that the sequence is Cauchy, and this establishes that the sequence is convergent. Limit Superior of a Sequence Given any sequence X(n) of real numbers, we define a new sequence h(m ) = Least Upper Bound of X(n ), n ≥ m ≡ sup X( n )

.

(11A4)

n≥m

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11A-2

EE603 Class Notes

10/24/2012

John Stensby

The term sup is an abbreviation for superior, or least upper bound. Given integer m, h(m) is the least upper bound of the set {X(m), X(m+1), … }. h(m) is a sequence of least upper bounds. Note that h(m) is a monotone decreasing sequence (h(m) ≥ h(m+1)). We discuss three possibilities for the behavior of h(m). First, sequence values h(m) may be +∞ for all m. This is true if, and only if, for all c ∈ R and integer n, there exists some k, k ≥ n, such that X(k) > c (no matter how “far out” you go, you can “go out further” and find “arbitrarily large positive” sequence elements). The second possibility is that, as m approaches infinity, h(m) may converge to a real number A ∈ R. In this case, for any ε > 0, there are infinitely many terms of the sequence that are greater than A - ε while only a finite number of terms are greater than A + ε. Finally, the third case is that h(m) may approach -∞ as index m approaches ∞. This is true if, and only if, the sequence X(n) itself approaches -∞ as n approaches ∞. The limit of h(m) is called the limit superior (the names lim sup and upper limit have been used in the literature) of the sequence X(n); the notation is ⎡ ⎤ A = lim sup X( n ) = limit h( m) = limit ⎢ sup X(n ) ⎥ , m →∞ m →∞ ⎣ n ≥ m n →∞ ⎦

(11A5)

where the limit over m is interpreted as an extended real number. (11A5) is referred to as the lim sup of the sequence. While the limit of X(n) may (or may not) exit, (11A5) always exists as an extended real number (the A = lim sup may be a finite real number, or it may be either +∞ or -∞; see the paragraph after (11A4)). Finally, the limit superior of a sequence and the limit superior of a sequence of sets (discussed in Appendix 11B) are similar notions. Alternate terminology exists in the literature. As defined by (11A4), h(m) is a monotone decreasing sequence. Hence, A = limit h( m ) = Greatest Lower Bound {h( m ), 0 ≤ m < ∞} ≡ inf h( m ) . m →∞

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m ≥0

(11A6)

11A-3

EE603 Class Notes

10/24/2012

John Stensby

Because of this, some authors write

A = lim sup X( n ) = inf n →∞

m ≥0

sup X( n ) .

(11A7)

n≥m

Limit Inferior of a Sequence

Given any sequence X(n), we define a new sequence g( m ) = Greatest Lower Bound of X(n ), n ≥ m ≡ inf X( n )

.

(11A8)

n≥m

The term inf is an abbreviation for inferior, or greatest lower bound. Given integer m, g(m) is the greatest lower bound of the set {X(m), X(m+1), … }. g(m) is a sequence of greatest lower bounds. Note that g(m) is a monotone increasing sequence (g(m) ≤ g(m+1)). We discuss three possibilities for the behavior of g(m). First, sequence values g(m) may be -∞ for all m. This is true if, and only if, for all c ∈ R and integer n, there exists some k, k ≥ n, such that X(k) < c (no matter how “far out” you go, you can “go out further” and find “arbitrarily large negative” sequence elements). The second possibility is that, as m approaches infinity, g(m) may converge to a real number A ∈ R. In this case there are infinitely many terms of the sequence that are less than A + ε while only a finite number of terms are less than A - ε, for any ε > 0. Finally, the third case is that g(m) may approach ∞ as m approaches ∞. This is true if, and only if, the sequence X(n) itself approaches ∞ as n approaches ∞. The limit of g(m) is called the limit inferior (lim inf and other names have been used in the literature) of the sequence X(n); the notation is A = lim inf X( n ) = limit g(m ) ≡ limit ⎡⎢ inf X( n ) ⎤⎥ , m →∞ m →∞ ⎣ n ≥ m ⎦ n →∞

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(11A9)

11A-4

EE603 Class Notes

10/24/2012

John Stensby

where the limit over m is interpreted as an extended real number. (11A9) is referred to as the lim inf of the sequence. While the limit of X(n) may (or may not) exit, (11A9) always exists as an extended real number (the A = lim inf may be a finite real number, or it may be either +∞ or -∞; see the paragraph after (11A8)). Finally, the limit inferior of a sequence and the limit inferior of a sequence of sets (discussed in Appendix 11B) are similar notions. Alternate terminology exists in the literature. As defined by (11A8), g(m) is a monotone increasing sequence. Hence,

A = limit g(m ) = Least Upper Bound {g(m), 0 ≤ m < ∞} ≡ sup g( m) m →∞

m ≥0

(11A10)

Because of this, some authors write

A = lim inf X( n ) = sup inf X( n ) n →∞

m ≥0 n ≥ m

(11A11)

The interpretations, in terms of ε, of lim inf and lim sup that are given above can be combined. Let X(n), n ≥ 0, be an arbitrary sequence of real numbers. For each ε > 0, there exists a finite integer n1 such that

A − ε < X( n ) < A + ε

(11A12)

for all n > n1. In words, for arbitrary ε > 0, all but a finite number of the X(n) fall between the numbers lim inf X(n) – ε and lim sup X(n) + ε. Relationships Involving lim inf and lim sup

Some simple relationships can be given between the limit superior and limit inferior of a sequence. Given any sequence X(n), we have the relationship

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11A-5

EE603 Class Notes

10/24/2012

A = lim inf X( n ) ≤ lim sup X( n ) = A . n →∞

John Stensby

(11A13)

n →∞

Also, given ε > 0, if an infinite number of X(n) values are less than A + ε, then an infinite number of −X(n) values greater than − A − ε. Likewise, if only a finite number of X(n) values are less than A − ε, then only a finite number of −X(n) are greater than − A + ε. Because of these facts, we can write − A ≡ − lim inf X( n ) = lim sup {− X( n )} n →∞

(11A14)

n →∞

(note: the sequence −X(n) is the negative of the sequence X(n)). Using similar reasoning, it is possible to argue that − A ≡ − lim sup X(n ) = lim inf {− X(n )} . n →∞

(11A15)

n →∞

Finally, the sequence X(n) converges to the extended real number A if, and only if, limit X(n ) = lim inf X(n ) = lim sup X( n ) n →∞

n →∞

n →∞

(11A16)

A=A= A .

The definition of the limit of a sequence of sets (considered in Appendix 11B) is the analog of (11A16). Example: Consider the sequence X( n ) =

1 + e-n/10 , n = 0, 2, 4, 6, " -n/10

= −1 − e

(11A17)

, n = 1, 3, 5, 7, "

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11A-6

EE603 Class Notes

10/24/2012

John Stensby

Clearly, this sequence does not converge. However, A = lim sup X(n) = 1

(11A18)

n →∞

since, for each ε > 0, there are only a finite number of terms larger than 1 + ε, but there are an infinite number of terms larger than 1 - ε. In a similar manner, we see that A = lim inf X(n) = −1

(11A19)

n →∞

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11A-7