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2bornot2b
 5 years ago
Why is the approximation theorem of supremum called so? I mean why is it named "approximation theorem"?
2bornot2b
 5 years ago
Why is the approximation theorem of supremum called so? I mean why is it named "approximation theorem"?

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JamesJ
 5 years ago
Best ResponseYou've already chosen the best response.3What result are you talking about?

2bornot2b
 5 years ago
Best ResponseYou've already chosen the best response.0I am talking about the following \[sup(a)\epsilon <x \le sup(a)\]

2bornot2b
 5 years ago
Best ResponseYou've already chosen the best response.0You know, I mean there exists an x etc etc

2bornot2b
 5 years ago
Best ResponseYou've already chosen the best response.0T.M Apostol calls is approximation theorem

JamesJ
 5 years ago
Best ResponseYou've already chosen the best response.3Give me the statement of the theorem. I don't know exactly what result you're talking about.

2bornot2b
 5 years ago
Best ResponseYou've already chosen the best response.0OK, I am stating it completely

2bornot2b
 5 years ago
Best ResponseYou've already chosen the best response.0Let 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 \[a<x \le b\]

JamesJ
 5 years ago
Best ResponseYou've already chosen the best response.3Yes, ok. It's saying you can get arbitrarily close the the sup.

2bornot2b
 5 years ago
Best ResponseYou've already chosen the best response.0They call it the "approximation property of Reals"

JamesJ
 5 years ago
Best ResponseYou've already chosen the best response.3For 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.

JamesJ
 5 years ago
Best ResponseYou've already chosen the best response.3sure, if you ask nicely. And I'll nearly always ignore them if they're not fresh.

JamesJ
 5 years ago
Best ResponseYou've already chosen the best response.3in other words, if I find it when I log on, then I'm almost certain not going to respond.
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