SOLVED
Let $A$ be a finite set and
\[B=\{ n \geq 1 : a\nmid n\textrm{ for all }a\in A\}.\]
Is it true that, for every $m>n\geq \max(A)$,
\[\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?\]
Cambie has observed that, if $A$ is the set of primes bounded above by $n$, and $m=2n$, then
\[\frac{\lvert B\cap [1,m]\rvert }{m}=\frac{\pi(2n)-\pi(n)+1}{2n}\sim \frac{1}{2\log n}\]
while
\[\frac{\lvert B\cap [1,n]\rvert}{n}=\frac{1}{n},\]
and hence the original question is false even with $2$ replaced by any constant $C$.
