Let $1 < a_1 < a_2 < a_3 <{} ...$ be a sequence of integers. For a subset $A \subset \Bbb Z$, denote by $d(A)$ its natural density (if it exists).
Is it true that $$ \lim_{N \to +\infty} d\Big( \bigcup_{i \leq N} a_i \Bbb Z \Big) = d\Big( \bigcup_{i = 1}^{\infty} a_i \Bbb Z \Big)$$?
The density of the finite union $\bigcup_{i \leq N} a_i \Bbb Z$ can be computed via the inclusion-exclusion principle, but it is not clear how it would relate to the infinite union.