511-Final-2004F - R m and a dierentiable function g : R R m...

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Economics 511 Professor John H. Nachbar Fall 2004 Final 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. 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. 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 and Y be vector spaces. If L : X Y is linear then dim( K ( L )) + dim( L ( X )) = dim( X ) . 4. Prove the following. Theorem. Let f : R n R m be differentiable, where n = + m . For x R n , let x λ denote the first coordinates of x and let x μ denote the last m coordinates of x . Similarly, let D λ f ( x ) denote the first columns of (the matrix representation of) Df ( x ) and let D μ f ( x ) denote the last m columns of Df ( x ) . Suppose that there is a y
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Unformatted text preview: R m and a dierentiable function g : R R m such that, for any x R , f ( x ,g ( x )) = y . For any x R , if D f ( x ) has full rank then Dg ( x ) =-[ D f ( x )]-1 D f ( x ) . 5. (a) Let X = { , 1 } . ( X consists of just two points.) Make X a metric space by giving it the standard Euclidean metric. Prove that the set { } is both open and closed in this metric space. (b) Let X = R n . The boundary of A R n , written A , is dened to be A = A A c . Prove that if both A and A c are nonempty then A 6 = . ( Hint. Fix two points a A and b A c . Show that there is a t such that x t = ta + (1-t ) b A .) (c) Again, let X = R n . Prove that if A and A c are both nonempty then A cannot be both open and closed....
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511-Final-2004F - R m and a dierentiable function g : R R m...

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