Mathematical Economics: Lecture 7
Transcript of Mathematical Economics: Lecture 7
Mathematical Economics:Lecture 7
Yu Ren
WISE, Xiamen University
October 15, 2012
math
Chapter 12: Limits and Open Sets
Outline
1 Chapter 12: Limits and Open Sets
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
New Section
Chapter 12: Limits andOpen Sets
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
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
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
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
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
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
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
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, . . .}
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Limit of Sequences
Limit of a sequenceIntuitive Definition: a number to which theentries of the sequences approacharbitrarily close. (How to define arbitrarily?)
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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}
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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}
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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, . . .
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Sequences of Real Numbers
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)
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
Sequences of Real Numbers
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)
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
Sequences of Real Numbers
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)
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
Sequences of Real Numbers
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
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
Sequences of Real Numbers
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
Yu Ren Mathematical Economics: Lecture 7
math
Chapter 12: Limits and Open Sets
Sequences of Real Numbers
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
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Sequences of Real Numbers
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Sequence in Rm
Definition: {xi}, xi ∈ Rm
Euclidean metric in Rm:
d(xi , xj) = ||xi − xj || =√(xi1 − xj1)2 + · · ·+ (xim − xjm)2
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Sequence in Rm
Definition: {xi}, xi ∈ Rm
Euclidean metric in Rm:
d(xi , xj) = ||xi − xj || =√(xi1 − xj1)2 + · · ·+ (xim − xjm)2
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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)
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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 .
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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 .
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Sequence in Rm
Accumulation Point: if for any positive ε, thereare infinitely elements in Bε(r)
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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}
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Properties
Theorem 12.7: open balls are open setsTheorem 12.8: any union of open sets isopen; the finite intersection of open sets isopen.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Properties
Theorem 12.7: open balls are open setsTheorem 12.8: any union of open sets isopen; the finite intersection of open sets isopen.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
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.
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Properties
Theorem 12.11: x ∈ clS if and only if thereis xn in S converging to x.Theorem 12.12: boundary = clS ∩ clSc
Yu Ren Mathematical Economics: Lecture 7
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Chapter 12: Limits and Open Sets
Properties
Theorem 12.11: x ∈ clS if and only if thereis xn in S converging to x.Theorem 12.12: boundary = clS ∩ clSc
Yu Ren Mathematical Economics: Lecture 7