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: d 1 and d 2 are equivalent metrics on X then the following hold: (i) A sequence { x n } X converges to x X in ( X,d 1 ) if and only if { x n } converges to x in ( X,d 2 ) . (ii) A subset S X is open under d 1 if and only if S is open under d 2 . 1 Exercise 3: Consider two metric spaces, ( X 1 ,d 1 ) and ( X 2 ,d 2 ) , and the product space, Z = X 1 X 2 . De ne the following function d on Z , for any x 1 ,x 1 X 1 and x 2 ,x 2 X 2 : d [( x 1 ,x 2 ) , ( x 1 ,x 2 )] = d 2 1 ( x 1 ,x 1 ) + d 2 2 ( x 2 ,x 2 ) / 1 / 2 Show that d is a metric. Note: d is called the product metric. 2...
View
Full
Document
 Fall '09
 Antoine
 Economics

Click to edit the document details