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2bornot2b

  • 4 years ago

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

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  1. JamesJ
    • 4 years ago
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    What result are you talking about?

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

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

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

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

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

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

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

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

  10. JamesJ
    • 4 years ago
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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.

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

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

  13. 2bornot2b
    • 4 years ago
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    OK, I understand.

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