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What result are you talking about?

I am talking about the following
\[sup(a)-\epsilon

You know, I mean there exists an x etc etc

T.M Apostol calls is approximation theorem

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

OK, I am stating it completely

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

They call it the "approximation property of Reals"

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

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

OK, I understand.