Let $A \in 2^{\mathbb{N}}$, for which $\mathbb{P}(A) := \lim_{n -> \infty}\frac{|A \cap [1,n]|}{n}$ exists, and $\mathbb{P}(A) \in \{0,1\}$.
So my question is, isn't it redundant to say $|A \cap [1,n]|$ as well as exists? Couldn't we just have said |A|? If A is finite then the intersection is still A and the limit will always be 0. If A is infinite, then the limit will always be 1, since A will be countably infinite.
Thank you!
Even if $A$ is infinite, it does not mean the limit is one. For example, consider $A$ only consists of powers of two. Then $\frac{|A\cap [1,n]|}{n} \approx \frac{\log(n)}{n} \to 0$. It's also possible to construct an $A$ where the limit does not exist.