Real Analysis with Economic Applications - Chapter B ppt

... countable for any m ∈ N. (b) Prov e or dispro ve: N ∞ is countable. Exercise 6.Let A and B be any sets such that A is countable, B is uncountable and A ⊆ B. Can B\ A = {x ∈ B : x/∈ A} be countable? ... irrational numbers countable? Exercise 7. H Let A and B be any sets such that A is countable and B is uncountable. Show that A ∪ B is uncountable. In fact, show that there is a b...

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... and B := co{y, z}. By Exercise 27, if an y v ector in B belongs to 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 ... 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 algebraic...

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... X,prove: (a) cl X (A)+cl X (B) ⊆ cl X (A + B) ; (b) cl X (cl X (A)+cl X (B) ) = cl X (A + B) ; (c) int X (A)+int X (B) ⊆ A +int X (B) ⊆ int X (A +B) , pro vided th at int X (B) = ∅. Exercise 12. H Show ... B) =co X (A)+co X (B) . Exercise 44. Prov e: For any subsets A and B of a metric linear space X, if both co X (A) and co X (B) are compact, then co X (A ∪ B) =co(co X (A) ∪ c...

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... {x}.)Ifeverymember of A isalsoamemberofB, then we say that A is a subset of B (also read “A is a set in B, ” or “A is contained in B ) an d write A ⊆ B (or B ⊇ A). Clearly, A = B holds iff both A ⊆ B and B ... while the former possibility arises only if a = b. But if a = b, then we have {{a}} =(a, b) =(a, b  )={{a}, {a, b  }} which holds only if {a} = {a, b  }, that i...

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... d bounded subset S of R n is compact. By boundedness, w e can find an ε > 0 such that S ⊆ N ε,R n (x) for some x ∈ S. Therefore, S m ust be a closed subset of a cube [a, b] n . 32 But [a, b] n is ... of [a, b] n no finite subset of which covers [a, b] n . (a)Bisect[a, b] n into 2 n equal cubes by planes parallel to its faces. At least one of these cubes is not covered by any finite...

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... previous one because it allo ws us to talk about the semicon tinuity of a function like ϕ • whic h ma y be extended real-< /b> valued (ev en w hen ϕ is real-< /b> valued). 20 Exercise 32. Let ϕ be any real < /b> function ... 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 Met...

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... c orrespondences b i : X −i ⇒ X i and b : X ⇒ X by b i (x −i ):=argmax{π i (x i ,x −i ):x i ∈ X i } and b( x): =b 1 (x −1 ) ×··· b m (x −m ), respective ly. (b i is calle d the best r esponse correspondence ... each i, H i : R 2 + → R is defined by H i (a, b) :=0if b =0, an d H i (a, b) := a b pf (b) − wa if b& gt;0, for som e p, w > 0 and strictly i n creasing f : R + → R...

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... agent is state-independent. 305 for all A, B ⊆ N with < /b> 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 < /b> A ∩ B = ∅. In the context of coalitional games, ... since d dθ  a θ + b θ  1 θ ≤ 0 for any a, b > 0, we have max{  a θ + b θ  1 θ : θ ≥ 1} = a + b. 309 Proposition 6. Let Y be a subset of a linear space X....

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... the r atio of blac k to r ed balls in the first urn — the so-called ambiguity of this urn — bothers the agen ts . This is a serious problem fo r the model en visaged by the Anscombe-Aumann Expected ... ened the axiomatic system of the Anscom be-Aumann theory to be able to cope with < /b> the Ellsberg P ar ad ox, but so what, you may say, w e could do so simply b y omitting the Independence...

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... let K be the class of all closed and convex sub sets A of S such that x ∈ A and f(A) ⊆ A. Define B := W K and C := co X (f (B) ∪ {x}). Then B = ∅ and f| B is a self-map on B. (b) B = C. (c) ζ (B) =ζ(f (B) ). ...  ∗∗∗∗FIG UR E J.1 A BO U T HER E ∗∗∗∗ 12 A famous problem of linear analysis < /b> (the so-called basis problem) was to determine if at least all separable Banach spaces...

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