{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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:
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern