Natural density ($d(S)$)

For $S\subseteq {\mathbb{Z}}_{\ge 0}$, its natural density, when it exists, is $d(S)={\lim }_{N\to \infty }|S\cap \{0,\dots ,N-1\}|/N$.

An increasing sequence of integers has full density when its range has natural density one. Convergence in density of a sequence means that, for each fixed tolerance, the exceptional indices have natural density zero.