{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

331mt2soln

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

This preview shows pages 1–2. Sign up to view the full content.

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 > 0 , there exists δ > 0 such that whenever d ( x, y ) < δ , u ( f ( x ) , f ( y )) < . Given δ > 0 , there exists N such that whenever n, m N , d ( a n , a m ) < δ . Then, given , there exists N such that whenever n, m N we have d ( a n , a m ) < δ so that u ( f ( a n ) , f ( a m )) < .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online