Yet another thread.. please bear with me... Dedekind question

Sorry i have no teacher so this is the only place i can get help. Btw i got another question about the Dedekind cut, i don't really understand what i need to do. It says : Using properties (A)-(H)

(In my book this is the basic properties of real numbers like commutative, associative,distributive laws.... the inverse of a number and the tricotomy property and transitive property ) of the real numbers and taking Dedekind theorem ( As stated in the previous question i posted) as given, show that every nonempty set U of real numbers that is bounded above has a supremum.

The hint is : Let T be the set of upper bounds of U and S be the set of real numbers that are not upper bounds of U.

What do i do? I mean i guess i can't use the completeness axiom. So how to i arrive at this result using only the basic properties of the real numbers and the Dedekind cut ? Btw Siclar Quote:

Originally Posted by

**siclar** It's still in reference to the OP so its not really a new question.

justchillin, I'm not sure what you are getting at with density. As far as the $\displaystyle \beta$ being unique, the idea behind these cuts is to construct the real numbers partitions of the rational numbers. In other words for each real $\displaystyle \beta$ the corresponding "cut" can be defined to be the rational numbers less than $\displaystyle \beta$. We can then define addition and multiplication and get something isomorphic to the real numbers as we know them. Then the uniqueness property now makes sense. We want each cut to uniquely represent each real number. I hope that clarifies things somewhat.

I don't understand what you mean when you say " these cuts is to construct the real numbers partitions of the rational numbers". What do you mean by partitions of rational numbers ? From what i understand we cut an interval of real numbers into two and then show that they are both bounded ( at the cut location) either from above or below . Is that what you mean ?