331mt2soln

331mt2soln - B U Department of Mathematics Fall 2005 Math...

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Unformatted text preview: B U Department of Mathematics Fall 2005 Math 331 - Real Analysis I, Second Midterm, 21/12/2005 1. (1) What does uniform convergence of a series of functions n =1 f n ( x ) mean? The uniform convergence of the sequence of its partial sums. 2. (1) Why do people sometimes require that n =1 f n ( x ) is to be uniformly convergent? It gives an answer to the question whether the limit function f ( x ) = n =1 f n ( x ) is continuous. 3. (1) Can a uniformly continuous function on R be uniformly convergent? Does not make sense. 4. (1) Which country was Abel from? Norway. 5. (8) Suppose ( M, d ) and ( N, u ) are two metric spaces and f : M N is uniformly continuous. Prove that a Cauchy sequence in M has its image under f a Cauchy sequence in N . Solution: Let { a n } M be Cauchy. We have: Given > , there exists > such that whenever d ( x, y ) < , u ( f ( x ) , f ( y )) < ....
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This note was uploaded on 05/04/2011 for the course MATH 331 taught by Professor Talinbudak during the Spring '11 term at BU.

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331mt2soln - B U Department of Mathematics Fall 2005 Math...

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