I was asked a question by one of my friends that seemed interesting however, I don't know how to start.
the question is that for any real number $a \in [0,1]$, show that there is a subsequence of number in $\mathbb{N}$, such that:
$$ \lim_{n\rightarrow \infty} \frac{\phi(m_n)}{m_n}=a. $$
I know that this limit is equivalent to show that for any $a \in [0,1]$, we want to show that: $$ \lim_{n\rightarrow \infty} \prod_{i=1} \left( 1- \frac{1}{p_{in}} \right)=a $$ $p_{in}$ is the $i^{th}$ prime divisor $m_n$.
