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The homework question (for a person I'm helping in Advanced Calculus) is prove the set $A=\{x \in \mathbb{Q} \colon x < 2\}$ is a Dedekind Cut (DC).

The third property of a DC is that it does not contain a greatest element. In mathese, that's $\forall a \in A,\ \exists b \in A\ \ni b>a$.

Set $A$ meets that requirement. If you say $a=\frac{19}{10}$, then I say $b=\frac{199}{100}$; if you say $a=\frac{199}{100}$, then I say $b=\frac{1999}{1000}$, etc. But how do I write that generally, so that it's a proof?

P.S. There is no tag for advanced-calculus, so I chose calculus.

Edit: I don't need a super formal proof. It just has to be general, for homework.

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    2-1/n is rational for every positve integer $n$.2012-02-03
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    For this I would have to first ask you, what is the definition of a Dedekind cut for you?2012-02-03
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    @AK: I'm not really concerned with the Dedekind cut definition. I just wanted to know how to write what I have in general terms.2012-02-04
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    @Jeff: I don't get it: your question is about how to write out carefully that a certain subset of $\mathbb{Q}$ is a Dedekind cut, but you're "not really concerned with the Dedekind cut definition". Que? Also, the most standard definition of a Dedekind cut (see e.g. http://en.wikipedia.org/wiki/Dedekind_cut) involves *two* subsets $A$ and $B$ of $\mathbb{Q}$. Thus according to many definitions what you have given is *not* a Dedekind cut. So please tell us what definition you're working with.2012-02-04
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    It is potentially offensive to refer to formal proof as "mathese".2012-02-04
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    I'm pretty sure the (not so un-common) definition employed is: nonempty, proper, downward closed, subset of $\mathbb{Q}$ without maximum. And OP had a problem with the last part, as stated in his question.2012-02-04
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    @PeteL.Clark: I guess I wasn't totally clear. What I was really wanted to ask was a way to write the paragraph that starts "Set $A$ meets that..." in a general form. The question wasn't specifically about Dedekind cuts. I thought the question was at least fairly clear - but clearly it's not clear. My bad.2012-02-04
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    @AM: Well, I haven't known anyone to get offended yet. If you are, please say so. I use the term mathese to emphasize that math is just a language - a rather concise language, to be sure - and that you can translate from English (e.g. word problems), or your chosen language, to "mathese".2012-02-04

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One way is: if I say $a$, then you say $\large b=\frac{a+2}{2}$. If $a<2$, then $b$ is always greater than $a$ because it's the number halfway between $a$ and $2$.

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    That's a good, general answer. I wonder if there isn't also a general way to write my idea in general terms?2012-02-04
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    I think the idea in your post, actually, works out to be almost the same thing. If you work out the formula for what you say in response to various values of a, it's b=(a+18)/20.2012-02-04