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204ps22006

# 204ps22006 - Economics 204 August 2006 Due Tuesday 8 August...

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Problem Set 2 Economics 204 - August 2006 Due Tuesday, 8 August in Lecture 1. Theorem 4 from the lim inf/lim sup handout: Let { x n } be a sequence of real numbers. Then lim x →∞ x n = x R ∪ {-∞ , ∞} if and only if lim inf x →∞ x n = lim sup x →∞ x n = x. Prove this theorem for the case that x is finite. 2. Using the definition of an open set, prove that (0 , 1) is an open subset of R (a) Under the usual absolute value metric. (b) Under the discrete metric. 3. Construct a sequence of real numbers (a) That is unbounded and has exactly three cluster points. (b) That is bounded and has infinitely many cluster points. 4. Prove that lim n →∞ ( n 2 + n - n ) = 1 / 2. 5. Closure, Boundary, Interior, etc. (a) Let A , B , denote subsets of a space X . Determine whether the following equations hold; if an equality fails, determine whether one of the inclusions or holds. 1. cl( A B ) = cl A cl B . 2. cl( A \ B ) = cl A \ cl B . (b) Find the boundary and the interior of each of the following subsets of R 2 : 1. A = { ( x, y ) | y = 0 } 2. B = { ( x, y ) | x > 0 and y = 0 } 3. C = A

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