Let $A$ be a subset of $\mathbb{N}$ and $(p_i)_i$ a strictly increasing sequence of positive integers. We define
$\overline{d}(A|(p_i)_i):=\limsup_n \frac{\left| A \cap \left\{ p_1,\cdots , p_n \right\} \right|}{n} $.
My question is: Let $A \subset \mathbb{N}$ and $(p_i)_i$, $(q_i)_i$ strictly increasing sequence of positive integers. If $\overline{d}(A|(p_i)_i)=1$ and $\overline{d}(\bigcup_{i \in \mathbb{N}}^{}\left\{ q_i \right\}|(p_i)_i)=1$, then we have $\overline{d}(A|(q_i)_i)=1$ ?
I couldn't prove it, Is there a counterexample?
Definitely not in general. Take $A$ to be a set whose density (the usual density in the integers) equals $\alpha$; take the $p_i$ to be precisely the elements of $A$, so that $\overline d(A|(p_i))=1$; and take $q_i=i$ for all $i$, so that $\overline d(\{q_1,q_2,\dots\}|(p_i))=1$. Then $\overline d(A|(q_i))=\alpha$, which can be any number in $[0,1]$.