{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

quiz3mat25 - Quiz 3 Solutions 1(5 pts Prove lim sup |sn | =...

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

View Full Document Right Arrow Icon
Quiz 3 Solutions 1. (5 pts) Prove lim sup | s n | = 0 iff lim s n = 0. Solution: see homework 4 solutions. 2. Suppose that lim sup s n = 1. (a) (5 pts) Construct a subsequence ( s n k ) which converges to 1. Use only definitions, no theorems. Solution: Given any > 0, there exist infinitely many elements s n k in the interval (1 - , 1+ ). (Make sure you can prove this statement. It’s pretty easy to do by contradiction.) Choose s n 0 = s 0 . Then inductively choose s n k to be one of the infinitely many elements in the interval (1 - 1 /k, 1 + 1 /k ), so that n k > n k - 1 (so we don’t pick the same sequence element twice). Then because | s n k - 1 | < 1 /k for all n k , s n k 1. (Again, make sure you can prove this. It has to do with the fact that given any > 0, you can find a k so that 1 /k < .) Note: nobody got this right. Maybe I should put it on the final. (b) (5 pts) Is it true that s n 1? Prove or give a counterexample. Solution: No, this is not true. For a counterexample, consider the se- quence defined by s n = ( - 1) n . 1
Image of page 1

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

View Full Document Right Arrow Icon
3. (10 pts) True or false. 2 pts each: 1 pt for correct answer, 1 pt for justification. (a) The set of subsequential limits of a bounded sequence is always non-empty.
Image of page 2
This is the end of the preview. Sign up to access the rest of the 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