Definition:Everywhere Dense/Real Numbers

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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 $S$.