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511-Midterm-2002F

# 511-Midterm-2002F - x ∈ X and any ε> there is a δ>...

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Economics 511 Professor John H. Nachbar Fall 2002 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. K X is closed iff, for any x X , if x is a limit point of K then x K . 2. Prove the following. Theorem. Let ( X, d X ) and ( Y, d Y ) be metric spaces. Fix y * Y and let f : X Y be the constant function defined by, for all x X , f ( x ) = y * . Then f is continuous. 3. Prove the following. Theorem. Let ( X, d X ) and ( Y, d Y ) be metric spaces. f : X Y is continuous iff, for any
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Unformatted text preview: x ∈ X and any ε > , there is a δ > such that, setting y = f ( x ) , N δ ( x ) ⊂ f-1 ( N ε ( y )) . 4. 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. 5. Let { x t } be a sequence in R and let E denote the set of subsequential limits of { x t } . (a) Provide an example in which E = [0 , 1]. (b) Provide an example in which E = R . (c) Prove that it is impossible to have E = (0 , 1)....
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