Consider $\theta_1$ ,$\theta_2$ $\in$ $\mathbb{R} \setminus \mathbb{Q}$ and linearly independent under $\mathbb{Q}$, i.e. if exists $\lambda_1$ , $\lambda_2$ $\in$ $\mathbb{Q}$ such that $\lambda_1 \theta_1 + \lambda_2 \theta_2 = 0$ $\Rightarrow$ $\lambda_1 = \lambda_2 = 0$, besides that let $D = \{(p + n \theta_1, q + n\theta_2)\hspace{0.1cm} ;\hspace{0.1cm} q, \hspace{0.1cm}p \hspace{0.1cm} \text{and} \hspace{0.1cm} n \hspace{0.1cm} \in \hspace{0.1cm} \mathbb{Z}\}$.
I want to prove that the set $D$ is dense in $\mathbb{R}^2$.
The only thing I managed to conclude was that the set $A = \{p+ n\theta \hspace{0.1cm};\hspace{0.1cm} n, \hspace{0.1cm} q \hspace{0.1cm} \in \hspace{0.1cm} \mathbb{Z}\}$ is dense in $\mathbb{R}$, but I can't apply this result to resolve my problem because both "entries" of the set $D$ have the same $n$.