Let $A$ be a subset of $\mathbb{Z}^{+}$ For any $n \in \mathbb{Z}^{+}$ put $A(n)=\{1,2,\ldots,n\} \cap A$

Define the upper asymptotic density $\overline{d}(A)$ of $A$ by

$$ \overline{d}(A) = \limsup_{n \rightarrow \infty} \frac{|A(n)|}{n} $$

$\overline{d}(A)$ is also known simply as the upper density of $A$
Similarly, we define $\underline{d}(A)$ the lower asymptotic density of $A$ by $$ \underline{d}(A) = \liminf_{n \rightarrow \infty} \frac{ |A(n)| }{n} $$ We say $A$ has asymptotic density $d(A)$ if $\underline{d}(A)=\overline{d}(A)$ in which case we put $d(A)=\overline{d}(A)$