Suppose that we have a sequence of finite sets $A_1, A_2, \ldots$, which partition $\mathbb{N}$. I am making no other assumptions on the $A_n$ - i.e. there could be any amount of interleaving between them. Now suppose we have $S\subset\mathbb{N}$. If $\lim_{n\rightarrow\infty} \frac{|S\cap A_n|}{|A_n|}=0$, does it follow that $S$ has a natural density of 0?
(And if so, while I'm at it, can 0 be replaced by other numbers? Natural density be replaced by upper, lower density? I mostly just care about the density 0 (equivalently density 1) case, though.)
EDIT: And if the statement is false, is there some sufficiently-little-interleaving condition on the $A_n$ I could assume that would make it true?