hw2 - d 1 and d 2 are equivalent metrics on X then the...

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ECON 831 Homework #2 due in class Oct. 1st Provide detailed calculations and justi cations to get full credit. Partial credit may be given. Exercise 1: Write the following mathematical statements with quanti ers and implication arrows. (i) De nition of an open set: A subset S of X is open in X if and only if, for each x S , there exists ² > 0 such that N ²,X ( x ) S . (ii) De nition of converging sequence: The sequence ( x m ) X converges to x X if and only if, for each ² > 0 , there exists a real number M ² such that d ( x m ,x ) < ² for all m M ² . Exercise 2: Consider the vector space X and two metrics on X , d 1 and d 2 . By de nition, d 1 and d 2 are equivalent if and only if there exists two real numbers a and b such that for all x and y X , we have: ( d 1 ( x,y ) ad 2 ( x,y ) d 2 ( x,y ) bd 1 ( x,y ) Show that if
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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...
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hw2 - d 1 and d 2 are equivalent metrics on X then the...

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