How to describe the set $$\{ 1 \leq n \leq N: \alpha_n \in(a,b)\}$$ when $(a,b) \subset [0,1)$ and you have following information:
a sequence of numbers $(\alpha_1,\alpha_2, \alpha_3,...)$, where $a_j \in [0,1)$, $j \in \mathbb{N}$, is equidistributed, if for every interval $(a,b) \subset [0,1)$ $$\lim_{N \rightarrow \infty} \frac{ \# \{1\leq n \leq N: \alpha_n \in (a,b) \}}{N}=b-a$$ holds, where $\# A$ is the cardinality of set $A$ (number of elements in the set).
I think $$A=\{ 1 \leq n \leq N: \alpha_n \in(a,b)\} = \{1,2,3,4, \dots, N \}\Rightarrow\#A = N\;,$$ but then how do you get $$\lim_{N \rightarrow \infty} \frac{ \# \{1\leq n \leq N: \alpha_n \in (a,b) \}}{N}=b-a\;?$$