This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 )) < ....
View
Full
Document
This note was uploaded on 05/04/2011 for the course MATH 331 taught by Professor Talinbudak during the Spring '11 term at BU.
 Spring '11
 talinbudak
 Math

Click to edit the document details