Q:

Is there any theory show that this claim :" for large $n$,$ 3/4$ of the positive integers less than $n$ are not divisible by $4$" is true?

I read   here in wikipedia for understanding distributions of square free , i find the following mathematical expression for large $n$ :" $ 3/4$ of the positive integers less than n are not divisible by $4$, $8/9$ of these numbers are not divisible by $9$ ?
My question here is :
Really i'm confused how the titled approximation is true and what are the mathematical theories in number theory or in probability   affirmed it's  truthness   ?
Thank you for any help 

A:

A statement of the form "for large $n$, fraction $p$ of the positive integers less than $n$ are in set $S$"
is to be interpreted as meaning
$$ \lim_{n \to \infty} \dfrac{| \{1,2,\ldots,n-1\} \cap S |}{n-1}  = p $$
In this case with $S$ the set of integers not divisible by $4$,
it's easy to prove (e.g. by induction) that $$|\{1,2,\ldots,n-1\} \cap S| = n - 1 - \lfloor (n-1)/4 \rfloor$$ so that
$$ \frac{3}{4}  = \frac{n-1 - (n-1)/4}{n-1} \le \frac{n-1 - \lfloor (n-1)/4 \rfloor}{n-1} \le \frac{n-1 - (n-1)/4 + 1}{n-1} = \frac{3}{4} + \frac{1}{n-1} $$
Then the  Squeeze Theorem shows the limit is $3/4$.

