Asymptotic density

Upper and lower asymptotic density

Let ${\displaystyle A}$ be a subset of the set of natural numbers ${\displaystyle \mathbb {N} =\{1,2,\ldots \}.}$ For any ${\displaystyle n\in \mathbb {N} }$, define ${\displaystyle A(n)}$ to be the intersection ${\displaystyle A(n)=\{1,2,\ldots ,n\}\cap A,}$ and let ${\displaystyle a(n)=|A(n)|}$ be the number of elements of ${\displaystyle A}$ less than or equal to ${\displaystyle n}$.