Quantcast

A community for students. Sign up today!

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

2bornot2b

  • 2 years ago

Why is the approximation theorem of supremum called so? I mean why is it named "approximation theorem"?

  • This Question is Closed
  1. JamesJ
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    What result are you talking about?

  2. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I am talking about the following \[sup(a)-\epsilon <x \le sup(a)\]

  3. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    You know, I mean there exists an x etc etc

  4. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    T.M Apostol calls is approximation theorem

  5. JamesJ
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Give me the statement of the theorem. I don't know exactly what result you're talking about.

  6. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    OK, I am stating it completely

  7. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    Let S be a nonempty set of real numbers with a supremum, say b=supS. Then for every a<b there is some x in S such that \[a<x \le b\]

  8. JamesJ
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Yes, ok. It's saying you can get arbitrarily close the the sup.

  9. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    They call it the "approximation property of Reals"

  10. JamesJ
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    For example, let S be the set of all rational numbers < sqrt(2). The sup of that set is clearly sqrt(2). The result is saying in this case for any rational number less than sqrt(2), you can find another rational number close to sqrt(2) ... so you can approximate sqrt(2) better and better if you want to/need to.

  11. JamesJ
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    sure, if you ask nicely. And I'll nearly always ignore them if they're not fresh.

  12. JamesJ
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    in other words, if I find it when I log on, then I'm almost certain not going to respond.

  13. 2bornot2b
    • 2 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    OK, I understand.

  14. 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.