How can I show this simple inequality?

Question

Let us consider $\mathbb{R}^n$ and $\lambda_1$, $\lambda_2 \in \mathbb{Z}^n$. For any $x\in 2^j \lambda_1 + [0,2^j)^n$ and $y\in 2^j \lambda_2 + [0,2^j)^n$, how can I show the following inequality,
$$ |x-y| \ge 2^{j-1}|\lambda_1 -\lambda_2|.$$ where $j$ is some integer. Moreover, can I replace $2^{j-1}$ by $2^j$? If then or not, why?

Answer 1

The inequality fails already for $n=1$ and $j=0$. Indeed, let $x=1$ and $y=\frac 23$. Then $\lambda_1=1$ and $\lambda_2=0$, so $|x-y|=\frac 13<\frac 12=\frac 12|\lambda_1-\lambda_2|$.