Mathematical Economics: Lecture 7

Mathematical Economics:Lecture 7

Yu Ren

WISE, Xiamen University

October 15, 2012

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Chapter 12: Limits and Open Sets

Outline

1 Chapter 12: Limits and Open Sets

Yu Ren Mathematical Economics: Lecture 7

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New Section

Chapter 12: Limits andOpen Sets


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Sequences of Real Numbers

A sequence of real numbers: a mappingfrom all natural numbers to real numbers.

There are infinite number of entries in asequenceMay not have an explicit function to describethe mapping


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ExampleExample 12.1 Some examples of a sequenceof real numbers are:

{1,2,3,4, . . .}{1, 1

2 ,13 ,

14 , . . .

}{1, 1

2 ,4,18 ,16, . . .

}{0,−1

2 ,23 ,−

34 ,

45 , . . .

}{−1,1,−1,1,−1, . . .}{2

1 ,32 ,

43 ,

54 , . . .

}{3.1,3.14,3.141,3.1415, . . .}{1,4,1,5,9, . . .}


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Limit of Sequences

Limit of a sequenceIntuitive Definition: a number to which theentries of the sequences approacharbitrarily close. (How to define arbitrarily?)


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Limit of Sequences

Limit of a sequenceMathematical Definition: for any {xn}, r isthe limit of this sequence if for any smallε > 0, ∃ N, s.t. for all n ≥ N, |xn − r | < ε.|xn − r | < ε⇐⇒ xn ∈ Iε(r)Iε(r) = {s ∈ R : |s − r | < ε}


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Limit of Sequences

Limit of a sequenceMathematical Definition: for any {xn}, r isthe limit of this sequence if for any smallε > 0, ∃ N, s.t. for all n ≥ N, |xn − r | < ε.|xn − r | < ε⇐⇒ xn ∈ Iε(r)Iε(r) = {s ∈ R : |s − r | < ε}


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Limit of Sequences: Example

Example 12.2 Here are three more sequenceswhich converge to 0:

1,0,12,0,

13,0, . . .

1,−12,13,−1

4, . . .

11,31,12,32,13,33,14, . . .


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Accumulation Point: if for any positive ε, thereare infinitely elements in Iε(r)

different with the limitonly need infinitely elements not all xn(n ≥ N)


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Properties of LimitsTheorem 12.1: A sequence can have atmost one limitTheorem 12.2 If xn → x and yn → y , thenxn + yn → x + yTheorem 12.3 If xn → x and yn → y , thenxn × yn → xy


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Subsequence: {yj} is a subsequence of {xi} if ∃an infinite increasing set of natural number {nj}s.t. y1 = xn1, y2 = xn2,y3 = xn3, for example{1,−1,1,−1, · · · } has two subsequences:{1,1, · · · } {−1,−1, · · · } and so on.


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Sequence in Rm

Definition: {xi}, xi ∈ Rm

Euclidean metric in Rm:

d(xi , xj) = ||xi − xj || =√(xi1 − xj1)2 + · · ·+ (xim − xjm)2


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Sequence in Rm

Definition: {xi}, xi ∈ Rm

Euclidean metric in Rm:

d(xi , xj) = ||xi − xj || =√(xi1 − xj1)2 + · · ·+ (xim − xjm)2


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Sequence in Rm

ε ball about r:Bε(r) ≡ {x ∈ Rm : ||x − r || < ε}{xi} is said to converge to the vector: forany ε > 0, ∃ N, s.t. for any n ≥ N,xn ∈ Bε(x)


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Sequence in Rm

PropertiesTheorem 12.5 xn → x if and only if xin → xifor all i = 1,2, · · · ,m.Theorem 12.6 xn → x , yn → y and cn → cthen cnxn + yn → cx + y .


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Sequence in Rm

PropertiesTheorem 12.5 xn → x if and only if xin → xifor all i = 1,2, · · · ,m.Theorem 12.6 xn → x , yn → y and cn → cthen cnxn + yn → cx + y .


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Sequence in Rm

Accumulation Point: if for any positive ε, thereare infinitely elements in Bε(r)


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Open Sets

Open sets: A set S in Rm is open if for eachx ∈ S, ∃ ε > 0 s.t. Bε(x) ⊂ S.Interior: S ⊆ Rm. The union of all open setscontained in S is called the interior of S,denoted by intS


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Open Sets

Open sets: A set S in Rm is open if for eachx ∈ S, ∃ ε > 0 s.t. Bε(x) ⊂ S.Interior: S ⊆ Rm. The union of all open setscontained in S is called the interior of S,denoted by intS


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Closed sets

Closed sets: A set S in Rm is closed ifwhenever {xn} is a convergent sequencecompletely contained in S, its limit is alsocontained in S.Closure: S ⊆ Rm. The intersection of allclosed sets containing S is called theclosure of S, denoted by clS.


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Closed sets

Closed sets: A set S in Rm is closed ifwhenever {xn} is a convergent sequencecompletely contained in S, its limit is alsocontained in S.Closure: S ⊆ Rm. The intersection of allclosed sets containing S is called theclosure of S, denoted by clS.


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Compact sets

Boundary: x is in the boundary of S if everyopen ball about x contains both points in Sand points in Sc.Bounded: a set S in Rm is bounded if ∃ anumber B st ||x || ≤ B for all x ∈ S.Compact sets: A set S in Rm is compact ifand only if it is both closed and bounded.


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Compact sets



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Compact sets



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Open Sets, Closed sets and compactsets

S = {x ∈ R2 : d(x ,1) ≤ 2}S = {x ∈ R2 : d(x ,1) < 2}S = {x ∈ R2 : 1 ≤ d(x ,1) ≤ 2}S = {x ∈ R2 : 1 < d(x ,1) ≤ 2}S = {x ∈ R2 : 1 < d(x ,1) < 2}


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Properties

Theorem 12.7: open balls are open setsTheorem 12.8: any union of open sets isopen; the finite intersection of open sets isopen.


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Properties

Theorem 12.7: open balls are open setsTheorem 12.8: any union of open sets isopen; the finite intersection of open sets isopen.


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Properties

Theorem 12.9: S is closed if and only ifRm − S is open.Theorem 12.10: any intersection of closedsets is closed; the finite union of closed setsis closed.


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Properties

Theorem 12.9: S is closed if and only ifRm − S is open.Theorem 12.10: any intersection of closedsets is closed; the finite union of closed setsis closed.


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Properties

Theorem 12.13: Any sequence containedin the closed and bounded interval [0,1]has a convergent subsequenceTheorem 12.14: Let C be a compactsubset in Rn and let {xn} be any sequencein C. Then, {xn} has a convergentsubsequence whose limit lies in C.


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Properties

Theorem 12.13: Any sequence containedin the closed and bounded interval [0,1]has a convergent subsequenceTheorem 12.14: Let C be a compactsubset in Rn and let {xn} be any sequencein C. Then, {xn} has a convergentsubsequence whose limit lies in C.


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Properties

Theorem 12.11: x ∈ clS if and only if thereis xn in S converging to x.Theorem 12.12: boundary = clS ∩ clSc


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Properties

Theorem 12.11: x ∈ clS if and only if thereis xn in S converging to x.Theorem 12.12: boundary = clS ∩ clSc


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Mathematical Economics: Lecture 7