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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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    @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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    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