Geometric condition for imaginary quadratic integer ring to be norm-Euclidean

Statement
Suppose $$\mathcal{O}$$ is an fact about::imaginary quadratic integer ring. Then, $$\mathcal{O}$$ is a fact about::norm-Euclidean ring of integers if and only if the following is true: for any point $$z \in \mathbb{C}$$, there exists a point $$w \in \mathcal{O}$$ such that:

$$\left| z - w \right| < 1$$.

Applications

 * Classification of norm-Euclidean imaginary quadratic integer rings

Facts used

 * 1) The norm in an imaginary quadratic integer ring is simply the square of the modulus as a complex number.
 * 2) The field of fractions of $$\mathcal{O}$$ is dense in $$\mathbb{C}$$.