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# hw1 - N ²,X x ⊆ X S 4(i Show that a subset of a metric...

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ECON 831 Homework #1 due in class Sept. 24th 1. Consider the following metric space ( X,d ) . Show that: | d ( x,y ) - d ( y,z ) | ≤ d ( x,z ) for any x,y,z X . 2. Consider the following two metric spaces with the same reference set, X , but di erent metrics: ( X,d 1 ) , and ( X,d 2 ) . (i) De ne the function δ = max { d 1 ,d 2 } . Is ( X,δ ) a metric space? Justify! (ii) De ne the function ρ = min { d 1 ,d 2 } . Is ( X,ρ ) a metric space? Justify! 3. Consider the proof of the following proposition (either refer to the notes or Ok, section 3.3, Prop. 5): "Any compact subset of a metric space X is closed and bounded." . At the end of the proof of closedness, justify the fact that:
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Unformatted text preview: N ²,X ( x ) ⊆ ( X \ S ) . 4. (i) Show that a subset of a metric space is bounded if it is totally bounded. (ii) Consider an in nite discrete space (say { x 1 ,x 2 ,x 3 , ···} ). Show that it is bounded but not bounded. ( Hint : consider the discrete metric) (iii) Show that in R n , the bounded subsets are exactly the totally bounded sets. 5. Consider the set of real numbers R and the discrete metric. (i) Show that the discrete metric on R is actually a metric. (ii) What are the subsets of R that are closed in the discrete metric? 1...
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