I'm having a little touble understanding application of the completeness axiom to certain subsets of real numbers. In a problem in a book (Fundamentals of Real Analysis by Haggarty), it asks you to show that the set S={a + b*sqrt(2) : a,b are rational} is not complete. As a hint, it tells you to show that sqrt(3) is not in the set. That's fine. What I don't understand is how it follows, from this last fact, that the set is not complete.(adsbygoogle = window.adsbygoogle || []).push({});

I know the Completeness axiom applies to bounded sets, so how can it be applied here. I tried following a similar approach to proving that Q is not complete, but I'm not sure if it's correct. It went something like this:

Take a subset of S such that a + b*sqrt(2) < sqrt(3), a,b rational. Then obviously the supremum (which is sqrt(3)) is not part of S and so it's not complete. Is this valid? It seems really trivial to me...

If anyone could enlighten me, that would be great.

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# Completeness Axiom

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