hw7 - X . Let Z ( f ) (the zero set of f ) be the set of...

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Homework 7 – Math 118A, Fall 2009 Due on Thursday, December 3rd, 2009 1. Investigate the convergence of a n where (a) a n = n + 1 - n n + 1 . (b) a n = s n + 1 - n n + 1 . 2. Let a n and b n converge, with b n > 0 for all n . Suppose that a n /b n L . Prove that lim N →∞ n = N a n n = N b n = L. 3. Prove that in R n all norms are equivalent, i.e., given two norms k · k and k · k 0 , there exist positive constants c < d such that c k x k ≤ k x k 0 d k x k . You can prove this by showing that every norm is equivalent to the Euclidean norm, and conclude the result from that. 4. Prove that if lim n →∞ x n = x, then lim ² 1 - (1 - ² ) X n =1 x n ² n = x. Prove that the converse is, in general, false by computing the second limit for x n = ( - 1) n . 5. Consider the Fibonacci sequence a 1 = a 2 = 1, and a n +1 = a n + a n - 1 , n 2 . Prove that lim n →∞ a n +1 a n exists, and ±nd the limit. 1
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2 6. Given a Cauchy sequence { x n } in a metric space X and a uniformly continuous function f of X into a metric space Y , prove that the se- quence { f ( x n ) } is a Cauchy sequence in Y . 7. Let f be a continuous real function on a metric space
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Unformatted text preview: X . Let Z ( f ) (the zero set of f ) be the set of all p X at which f ( p ) = 0. Prove that Z ( f ) is closed. 8. Let f and g be continuous mappings of a metric space X into a metric space Y , and let E be a dense subset of X . Prove that f ( E ) is dense in f ( X ). If g ( p ) = f ( p ) for all p E , prove that g ( p ) = f ( p ) for all p X . 9. Let I = [ a, b ] a closed interval. Suppose f : I I is continuous. Prove that there must be at least one point x [ a, b ] such that f ( x ) = x . 10. Let X be a metric space, Y a complete metric space, and E X a dense set. Suppose that f : E Y is uniformly continuous. Prove that there exists a unique continuous extension of f to the whole space X , i.e., ! g : X Y continuous such that f ( x ) = g ( x ) for all x E ....
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hw7 - X . Let Z ( f ) (the zero set of f ) be the set of...

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