# The Set of integer multiples of two irrational numbers is dense in reals?

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?

• $A$ 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 $\alpha,\beta$.
• Do you mean $x < 0$? – amsmath Oct 12 '17 at 20:43