Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

swissgirl

  • one year ago

Show that any non-empty finite set S ⊂ ℝ contains both its supremum and infimum. (Hint: use induction)

  • This Question is Closed
  1. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @abb0t

  2. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    To be honest I really didnt study the chapter fully yet I was just looking for a tutor so hopefully I wont be lost lol

  3. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \[S \subseteq R\] such that |S| = n Proceed by induction on n. If n = 1, then S = {a} for some \[a \in R\]. Then, sup S = a. Assume that all subsets of R of order k contain their respective suprema. Let S be a subset of R in order of k+1. Label the elements of S as: \[\left\{ a_1, a_2...a_{k+1} \right\}\] Let \[T = \left\{ a_1, a_2...a_k \right\} \] By the inductive assumption: \[\sup T= a_{i_o}\] for some \[1 \le i_o \le k\] and \[\sup(S-T) = a_{k+1}\] Thus clamining case 1: \[a_{i_o} \le a_{k+1}\] since \[a_{i_o} = \sup T\] and \[a_{i_o} \le a_{k+1}\], you know that for all \[1 \le i_o \le k, a_i \le a_{i_o} \le a_{k+1}\] thus \[\sup S = a_{k+1}\] Case 2: \[a_{i_o} \ge a_{k+1}\] since \[a_{i_o} = \sup T\] and \[a_{i_o} \ge a_{k+1} \] you know that for all \[1 \le i \le k+1, a_i \le a_{i_o}\] thus \[\sup S = a_{i_o}\] Thus, S contains its supremum

  4. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Give me time to read it. Thanks :)

  5. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Wow this is great :)

  6. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I didnt follow case 2 though

  7. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ummm Also werent we supposed to prove that it contains its supremum and infimum

  8. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Like I followed what u did but i am just confused why we needed to show case 2

  9. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Well supremem is the upper bound, and infimum is the lower bound. I just used the definitions.

  10. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    No like how did u show S contains the infimum

  11. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Case 2.

  12. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I showed4 supremum.

  13. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Ohhhhhh ok I thought i was going senile

  14. abb0t
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    :) nah don't say that.

  15. swissgirl
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    lol Thankssssssss. THIS WAS AWESOME. U R REALLY CLEARRRRRRRR

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

    • Attachments:

Ask your own question

Ask a Question
Find more explanations on OpenStudy

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.