Can we Use Exclusion-Inclusion Principle for a Countably Infinite Set? A popular question in Mathematics is that: What percent of Counting Numbers are divisible by 3? 
Without thinking twice, one might answer 33 percent.
An Extension of this is the question: What percentage of Counting Numbers are divisible by 3 or 5?
To solve this problem, others usually used Inclusion-Exclusion Principle.
However, references always indicate that Exclusion-Inclusion Principle must be done in a finite set. 
My question is: Is Exclusion-Inclusion Principle valid for a countably infinite set? Thank you so much for your answers. 

For $n\in \mathbb N$ let $P(n)$ be the number of $m\in \mathbb N$ that do not exceed $n$ and are NOT divisible by $3$ nor by $5.$
Let $15a+b=n$ where $0\leq a\in \mathbb N$ and $b\in  \mathbb Z$ with $0\leq b\leq 14.$ It is, I hope, obvious that $$P(15a)=(1-1/3)(1-1/5)15a=(8/15)15a\geq (8/15)(n-14).$$ And that $$P(15a+15)=(1-1/3)(1-1/5)P(15a+15)=(8/15)(15a+15)\leq (8/15)(n+15).$$ Since $P(15a)\leq P(n)\leq P(15a+15)$ we have $$(8/15)(1-14/n)\leq P(n)/n\leq (8/15)(1+15/n).$$ So $\lim_{n\to \infty}(1-P(n)/n)=7/15.$
