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I have a problem that I am seriously stuck on. I'm not sure what to do I've seen similar proofs online with the least positive rational number but this is apparently different and I'm not sure why.

Prove the statement “There is no smallest rational number greater than 2” by contradiction.

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    $\begingroup$ You don't have a proof, you have a problem. It's unnecessary to emphasize "proof" so much in tags and title: pretty much everything in mathematics is about proofs. $\endgroup$
    – user147263
    Nov 8, 2015 at 3:20
  • $\begingroup$ I agree but this requires us to have this as a proof, in proof format, and logically proven to be true or false. $\endgroup$
    – mm19
    Nov 8, 2015 at 3:23
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    $\begingroup$ Everything ought to be logically proven to be true or false, when possible. The requirement of a proof may be unusual to you, but it's not unusual in mathematics. $\endgroup$
    – user147263
    Nov 8, 2015 at 3:25

1 Answer 1

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Suppose there was a smallest rational number greater than $2$. Call it $k =p/q$.

Then consider $ k' = \frac{2+k}{2}$. This is a number bigger than $2$ and less than $k$. Also $k'$ is rational. Therefore there is no smallest rational number greater than $2$.

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  • $\begingroup$ Thank you I understand what your saying. I think I'll be able to write something up now. $\endgroup$
    – mm19
    Nov 8, 2015 at 3:31

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