Real Analysis with Economic Applications - Figures docx

... contraction with the unique fixed point ¯x. f is not a contraction, and it has no fi x ed points. Figure C.4 sin x cos x π 3π 2 1 −1 ln x 1 e x 1 Figure A.3 Figure D.7 1 2 7 8 1 f 1 f 2 f 3 -1 1 x 1 x 2 x 1 x 2 = ... convex. S co(S) S is not convex. Figure G.1 f 1 f 2 f 3 f 4 1 1 Figure D.6 0 0 0 0 0 0 0 0 0 0 Non-convex cones in R 2 Convex cones in R 2 Sets in R 2 which are not cones Figu...

Ngày tải lên: 04/07/2014, 10:20

... on int R 2 (S). 35 5.1 Finite-Horizon Var iational Problems We begin with laying out, in the abstract, the class of variational problem s that we shall be concerned with here. Let x 0 and τ be real n u mbers, ... for f ree-end smooth variational problems, prov ides a case in poin t. Corollary 5. Le t [Ω, ϕ,x 0 , τ , F] be a free-end sm ooth variational problem. If (29) holds for some f...

Ngày tải lên: 04/07/2014, 10:20

... corollary. The Bolzano-Weierstrass Theorem. 34 Ev ery bounded real sequence has a con- v ergen t subsequence. Exercise 37. Show that every unbounded real sequence has a subsequence that di- v erges to ... e difference-quotient map Q f,x : I\{x} → R by Q f,x (t):= f(t)−f(x) t−x . If the right-limit of this map at x exists as a real number, that is, Q f,x (x+) ∈ R, then f is said to be rig...

Ngày tải lên: 04/07/2014, 10:20

... to be a well-ordering if ev ery nonempty subset of X has a -minimum. In this case X is said to be w e ll-ordered by ,and(X, ) is called a well-ord e red s et,orshortlyawoset. Well-ordered sets ... Principle, it contains a ⊇-maximal loset, say (Z i , ). By -densene ss of Y , Z i has neither a -maxim um nor a -minimum. Moreove r, by its maximalit y, it i s -d en s e in itself. By Cor .....

Ngày tải lên: 04/07/2014, 10:20

... all bounded real sequences, th at is,  ∞ := {(x m ) ∈ R ∞ :sup{|x m | : m ∈ N} < ∞} . It is implicitly understood that this set i s e ndowed with the metric d ∞ :  ∞ × ∞ → R + with d ∞ ((x m ) ... R ∞ since the mem bers of such a space are real sequences that are either bounde d or that satisfy some form of a summability condition (that ensures that d p is real- valued). Indeed...

Ngày tải lên: 04/07/2014, 10:20

... semicon tinuity of a function like ϕ • whic h ma y be extended real- valued (ev en w hen ϕ is real- valued). 20 Exercise 32. Let ϕ be any real function defined on a metric space X. Pro ve: (a) ϕ • ≤ ... prove this approximation-by-continuous-functions theorem, but you might want to try it out for yourself in the case where X is a compact interval. 176 3.2 The Local-to-Global Method Since a...

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... open subset O of R n + with B(p, ι) ∩ O = ∅. To derive a con trad iction, suppose that for every m ∈ N (howev er large), there exists an (n +1)-vector (p m , ι m ) within 1 m -neighborh ood of (p, ... R n .IfΓ is a convex-valued self-co rres pondenc e on X that has a closed graph, then Γ has a fix ed point, tha t is, there exists an x ∈ X with x ∈ Γ(x). 20 Sadly, Shizuo Kakutani (191 1-2...

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... uniqueness claims made in t he Anscombe-Aumann Ex- pected Utility Theorem. ∗ Exercise 46. Obtain the counterpart of the Anscombe-Aumann Expected Utility Theorem (with modifying some of the axioms ... the risk p refer- ences of the agent is state-independent. 305 for all A, B ⊆ N with A ∩ B = ∅. The capacit y v is said to be additive if v(A ∪ B)=v(A)+v(B) for all A, B ⊆ N with A ∩ B = ∅....

Ngày tải lên: 04/07/2014, 10:20

... al-int X (S), then B\{y, z } ⊆ al-int X (S). (Why?) But this wou ld imply that x ∈ al-int X (S), a contradiction. Th us al-int X (S)∩B = ∅. In turn, this means that al- int X (A) = ∅ an d al-int X (A) ... S (in X), and is denoted by al-cl X (S). (Note. al-c l X (∅)=∅.)Ifal-cl X (S) ⊆ S, we say that S is algebr a ically closed in X. F in ally, the set al-cl X (S)\al-int X (S) is called the al...

Ngày tải lên: 04/07/2014, 10:20

... nscombe-Aumann setup, the prefer- enc e relation  of an ind iv idu al is defined on H Ω,X . If this preference relation is com- plete and it satisfies the Independence, Cont inuity, No-Tr iv iality, ... conclude the section with the proof o f the Gilboa-Schm eidler Theorem. T his is a “deep” result, but on e which is not all that hard to establish once o ne realizes that convex analysis li...

Ngày tải lên: 04/07/2014, 10:20