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Suppose I have $0 < N - \varepsilon < q$ where $N$ is some large real number and $\varepsilon$ is much smaller than $N$. Can I conclude that $0 < N - \varepsilon < N < q$?

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    @RossMillikan: when someone has only asked 5 questions, I don't think it's *that* unlikely none of the answers are acceptable...2012-09-17

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No. You have $0, but it is entirely possible that $q$ lies between $N-\epsilon$ and $N$.

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    Never mind. Thanks.2012-09-17
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Think about $0 < 100000 - 2 < 99999$ ...