About density in $\mathbb{R}^{3}$

Suppose that $\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\in\mathbb{R}^{+}\setminus\mathbb{Q}$.

Does anyone have any suggestions to prove the following$$\overline{\mathbb{R}\left(1,\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\right)+\mathbb{Z}^{3}}=\mathbb{R}^{3} \hspace{1cm}?$$

I tried using sequences, I cannot obtain a sequence of integers that converges and satisfies the convergence in each component.

Here

$\mathbb{R}^{3}=\{ (x_{1}, x_{2}, x_{3}) : x_{i}\in\mathbb{R},\hspace{0.1cm}\forall j=1,2,3 \}$.

$\mathbb{R}\left(1,\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\right)=\{ r\left(1,\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\right)=\left(r,r\frac{\lambda_2}{\lambda_1}, r\frac{\lambda_3}{\lambda_1}\right) : r\in\mathbb{R}\}$ in other words the spanned (over $\mathbb{R}$) of $\left(1,\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\right)$.

Thanks