TITLE: How to find the limit points of the set $\{\ a+\alpha\ b \mid a,b \in \mathbb Z, \ \alpha\ \text{is a fixed irrational number} \}$
QUESTION [1 upvotes]: How to find the limit points of the set $\{\ a+\alpha\ b \mid a,b \in \mathbb Z, \ \alpha\ \text{is a fixed irrational number} \}$
limit point: A point $x$ is said to be a limit point of a non empty set $X,$ if every deleted neighborhood of $x$ contains at least a point in $X.$
Equivalently, There is a sequence $(x_n)$ in $X$ such that $(x_n)\to x.$
Using this definition, how can I find the limit points of the set  $\{\ a+\alpha\ b \mid a,b \in \mathbb Z, \ \alpha\ \text{is a fixed irrational number} \}$ ?
Hints will be greatly appreciated...

REPLY [0 votes]: I'm going to show that the set $A=\{n+m\alpha\,|\,m,n\in\mathbb Z\}$ is dense in $\mathbb R$. For this I will show that $A$ intersects every open interval in the real line.
Suppose that $(a,b)$ is an arbitrary non-empty interval in the real line. Then by Dirichlet's approximation theorem there exists integers $m,n$ such that
$$
0<|m\alpha+n|<b-a
$$
Now for some natural integer $N$ we must have $a<N|m\alpha+n|<b$, but $N|m\alpha+n|\in A$ as we wanted.
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