Set of Reciprocals of Positive Integers is Nowhere Dense in Reals

Theorem
Let $N$ be the set defined as:
 * $N := \left\{{\dfrac 1 n: n \in \Z_{>0}}\right\}$

where $\Z_{>0}$ is the set of (strictly) positive integers.

Then $N$ is nowhere dense in the set of real numbers $\R$ considered as a metric space.