Let $\alpha , \beta $ be the linearly independent irrational numbers over $\mathbb Q$ with $\alpha > \beta > 0 $ , and $\mathrm A=\{n\alpha-m\beta \mid n,m \text{ are nonnegative integers} \}$

How to prove that $\mathrm A$ is dense in $\mathbb R$ ? Is it true?


Note that

  • $A$ is closed under addition
  • $\inf A=-\infty$
  • $\sup \{\,x\in A\mid x>0\,\}=0$.

The first is trivial, the second follows from $\beta>0$, and only the third involves the $\Bbb Q$-linear independence of $\alpha,\beta$.

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  • $\begingroup$ The third does not make sense $\endgroup$ – amsmath Oct 12 '17 at 20:25
  • $\begingroup$ The third is wrong.. $\endgroup$ – merow Oct 12 '17 at 20:38
  • $\begingroup$ Do you mean $x < 0$? $\endgroup$ – amsmath Oct 12 '17 at 20:43
  • $\begingroup$ I'm not sure why you rejected my edit to the third bullet point: it makes no sense as written. $\endgroup$ – Steve D Oct 13 '17 at 4:24

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