For any integer $n>0$ there is always a prime $p$ with $q_n\leq{p}\leq{3n}$ or $3n\leq{p}\leq{q_n}$ , where $q_n$ is the $n$-th prime.

Question

I checked it for all the prime $\leq{100}$, the results seem very obvious though, so I would love a proof or two and any correction and opinion. Thanks in advance.

Answer 1

We know that: $$q_n\sim n*log\ n$$ $log\ n$ grows, so from some point on, there will be so much space inbetween $3n$ and $q_n$ that it will work for most numbers. You can take any number instead of $3$ and this will still be true.