I am reading a paper where the author states that a map with the following property exists.
Let $\Omega\subset B(0,R)\subset \mathbb{R}^d$ for $R>0$ and $n\geq 1.$ Define $p_n$ to be a function that goes from $\Omega$ to a grid of at most $(2Rn)^d$ points with the property that $$|p_n(x) - x|\leq 1/n.$$
I am trying to understand why such a map exists and what would it look like explicitly.
Here are some of my thoughts. At first, I guessed that $p_n(x) = [nx]/n$ where $[\cdot ]$ denotes the nearest lattice point in $\mathbb{R}^d.$ The this map will satisfy the inequality since $|[x]-x|\leq 1$ for all $x\in \mathbb{R}^d$ however I am not sure if this map works. Any suggestions will be much appreciated.
