A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • 4 years ago

How can I tell if the sequence\[\left(1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},...\right)\]is compact or not?

  • This Question is Closed
  1. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I noticed that the sequence converges to \(1\), which is in the set. I also noticed that every subsequence will also converge to \(1\). To me, it seems the sequence is compact.

  2. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yes i think it is compact, which i believe means any subsequence \[\{x_n\}_{n\geq 1}\] converges to something in the set

  3. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    of course it is always possible that your subsequence only has a finite number of distinct elements, but that is ok because if such is the case then at least one must occur infinitely often and you can define the subsequence \[\{x_{n_k}\}=x\] for such an x

  4. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    if i recall correcty ( it has been a while) sequence is compact if every subsequence contains a further subsequence that is convergent

  5. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    That makes sense. :) Thank you

  6. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    yw but you still have to say something. for example if you have a subsequence \[\{x_n\}\] you have to explain why it would have a further subsequence \[\{x_{n_k}\}\] that converges

  7. anonymous
    • 4 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    so for example you could make \[\{x_{n_k}\}\] be something like \[x_k \in \{x_n\} \cap (1-\frac{1}{k},1+\frac{1}{k})\]man that took a long time to type

  8. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.