Question

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

Answer 1

What result are you talking about?

Answer 2

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

Answer 3

You know, I mean there exists an x etc etc

Answer 4

T.M Apostol calls is approximation theorem

Answer 5

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

Answer 6

OK, I am stating it completely

Answer 7

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
\[a<x \le b\]

Answer 8

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

Answer 9

They call it the "approximation property of Reals"

Answer 10

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.

Answer 11

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

Answer 12

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

Answer 13

OK, I understand.