# Definition:Everywhere Dense/Real Numbers

Let $S$ be a subset of the real numbers.
Then $S$ is (everywhere) dense in $\R$ if and only if:
$\forall x \in \R : \forall \epsilon \in \R : \epsilon > 0 : \exists s\in S : x-\epsilon < s < x + \epsilon$.
That is, if and only if in every neighborhood of every real number lies an element of $S$.