$ 1+\mathsf{d}^\star(B) \le \mathsf{d}^\star(A\cup B)+\mathsf{d}^\star(B\cup C) $ with $A\cup B\cup C=\bf{N}^+$

Question

For each subset of positive integers $X\subseteq \bf N^+$ define the upper asymptotic density as $$ \mathsf{d}^\star(X)=\limsup_{n\to \infty} \frac{|X\cap [1,n]|}{n}. $$

Problem: Let $A,B,C$ be a partition of the positive integers $\bf N^+$. Is it true that $$ 1+\mathsf{d}^\star(B) \le \mathsf{d}^\star(A\cup B)+\mathsf{d}^\star(B\cup C)? $$

Answer 1

Let $d_n(X) = \frac{|X\cap[1,n]|}{n}$. Then: $$\begin{eqnarray*}1 + d_n(B) = \frac{n + |B\cap[1,n]|}{n} &=& \frac{|(A\cup B)\cap[1,n]| + |(B\cup C)\cap[1,n]|}{n}\\[0.2cm] &=& d_n(A\cup B)+d_n(B\cup C)\end{eqnarray*} $$ and hence: $$\begin{eqnarray*}1 + d^\star(B) = \limsup\limits_{n\rightarrow\infty}\left(1 + d_n(B)\right) &\color{red}{\le}&\limsup\limits_{n\rightarrow\infty}{d_n(A\cup B)} + \limsup\limits_{n\rightarrow\infty}{d_n(B\cup C)}\\[0.2cm] &=& d^\star(A\cup B) + d^\star(B\cup C). \end{eqnarray*}$$