4
Let $\mathscr{F}$ be a free ultrafilter on $\mathbf{N}$ and, for each $A\subseteq \mathbf{N}$ and $n \in \mathbf{N}$, define
$$
d_n(A):=\frac{|A\cap [1,n]|}{n}.
$$
Question. Considering $\mathcal{P}(\mathbf{N})=\{0,1\}^{\mathbf{N}}$, is it true that the set
$$
\{A\subseteq \mathbf{N}: \math...