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2bornot2b Group Title

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

  • 2 years ago
  • 2 years ago

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  1. JamesJ Group Title
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    What result are you talking about?

    • 2 years ago
  2. 2bornot2b Group Title
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    I am talking about the following \[sup(a)-\epsilon <x \le sup(a)\]

    • 2 years ago
  3. 2bornot2b Group Title
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    You know, I mean there exists an x etc etc

    • 2 years ago
  4. 2bornot2b Group Title
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    T.M Apostol calls is approximation theorem

    • 2 years ago
  5. JamesJ Group Title
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    Give me the statement of the theorem. I don't know exactly what result you're talking about.

    • 2 years ago
  6. 2bornot2b Group Title
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    OK, I am stating it completely

    • 2 years ago
  7. 2bornot2b Group Title
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    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 \[a<x \le b\]

    • 2 years ago
  8. JamesJ Group Title
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    Yes, ok. It's saying you can get arbitrarily close the the sup.

    • 2 years ago
  9. 2bornot2b Group Title
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    They call it the "approximation property of Reals"

    • 2 years ago
  10. JamesJ Group Title
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    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.

    • 2 years ago
  11. JamesJ Group Title
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    sure, if you ask nicely. And I'll nearly always ignore them if they're not fresh.

    • 2 years ago
  12. JamesJ Group Title
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    in other words, if I find it when I log on, then I'm almost certain not going to respond.

    • 2 years ago
  13. 2bornot2b Group Title
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    OK, I understand.

    • 2 years ago
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