I want to show that the sequence

$$ x_n=\left(\frac{3+\sqrt{5}}{2}\right)^n\cdot\begin{pmatrix}\frac{1+\sqrt{5}}{2}\\ 1\end{pmatrix}\,\mathrm{mod}\,1\in\mathbb{R}^2 $$ is dense in the unit square $\lbrack 0,1\rbrack^2$ (The $\mathrm{mod}\,1$ operator is taken componentwise). Unfortunately I'm not able to show this. Any kind of help is appreciated.