I am trying to solve this problem. Let us say that a number $n$ is smooth if it does not have prime factors bigger than $\sqrt{n}$. Let $f$ be the function that counts the smooth numbers between $1$ and $x$. How can I prove that there exist two constants $c_1>0$ and $c_2<1$ such that for every $x$ sufficiently big $c_1 x < f(x) < c_2 x$?
I would appreciate if you can give me just a small hint. Thank you in advance.