511-Midterm-2004F - if lim x t = x then f x ≤ lim inf f x...

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Economics 511 Professor John H. Nachbar Fall 2004 Midterm You have one and a half hours. Write your answers clearly, with good penmanship and good syntax. A “correct” but unintelligible answer is a wrong answer. 1. Prove the following. Theorem. Let ( X, d ) be a metric space. For any x X and any ε > 0 , N ε ( x ) is open. 2. Prove the following. Theorem. The series x i is convergent iff for every ε > 0 , there is a number T such that, for all t > s > T , t X i = s +1 x i < ε. 3. Prove the following. Theorem. Let ( X, d x ) and ( Y, d Y ) be metric spaces. Let f : X Y be continuous. For any compact set A X , f ( A ) is compact. 4. Prove the following. Theorem. Let X and Y be vector spaces. Let L : X Y be linear. Then L ( X ) is a vector subspace of Y . 5. Consider a function f : R n R . Say that f is lower semicontinuous iff for any y R , { x R n : f ( x ) y } is closed. (a) Show, using the definition, that the function f : R R defined by f ( x ) = ( 1 if x 0 , 0 otherwise , is not lower semicontinuous.
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(b) Prove that f : R n R is lower semicontinuous iff for any x * R n and any sequence { x t } , if lim
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Unformatted text preview: , if lim x t = x * then f ( x * ) ≤ lim inf f ( x t ) . Note also that one possible sequence of x t is x t = x * for all t . Informally, a function is lower semicontinuous iff the function only jumps down, if it jumps at all. 1 (c) The epigraph of a function f : R n → R is the set { ( x,y ) ∈ R n +1 : y ≥ f ( x ) } . Informally, the epigraph of f is the set of points lying on or above the graph of f . Prove that f is lower semicontinuous iff the epigraph of f is closed. 1 In case you’ve forgotten the definition of lim inf: let { x t } be a sequence in R n and let E ⊂ R ∪ {-∞ , ∞} be the set of subsequential limits of { x t } . Then lim inf x t = inf E . One can show (the proof is not difficult but I do not require you to provide it) that there is a subsequence { x t k } such that lim x t k = lim inf x t ....
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