If $A\subset\mathbb N$ does not contain any two integers such that one divides the other, must $A$ have density $0$?

Question

Suppose we have a set $A\subset\{1,2,3,\ldots\}$ such that there do not exist $m,n\in A$ where $m\mid n$. Does it follow that $A$ has natural density $0$?

(Note: The definition of the natural density of $A$ is $\delta(A):=\lim\limits_{n\to\infty} \dfrac{|A\cap\{1,2,\ldots,n\}|}{n}$.)