2bornot2b 3 years ago Why is the approximation theorem of supremum called so? I mean why is it named "approximation theorem"?

1. JamesJ

What result are you talking about?

2. 2bornot2b

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

3. 2bornot2b

You know, I mean there exists an x etc etc

4. 2bornot2b

T.M Apostol calls is approximation theorem

5. JamesJ

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

6. 2bornot2b

OK, I am stating it completely

7. 2bornot2b

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

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

9. 2bornot2b

They call it the "approximation property of Reals"

10. JamesJ

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

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

12. JamesJ

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

13. 2bornot2b

OK, I understand.