suppose that $\phi(n)$ is Euler function. prove that, $\{\frac{\phi (n)}{n}\}_{n \in \Bbb N}$ is dense in $[0,1]$
(if $A_n=\{1 \leq m \leq n \mid m \in \Bbb N ; \gcd(n,m)=1\}$ then $\phi(n)=|A_n|$)
I think : If $p$ is prime then $\phi(p)=p-1$. for any $ \varepsilon > 0 $ there is a prime number $p$ such that $1-\varepsilon \leq \frac{p-1}{p} \leq 1$. for other elements i don't know what can i do.