Compare nth prime number to 3n

Question

Prove that $\forall n \in N, n \ge 12: P_{n} > 3n$ where $P_{n}$ is the $n$th prime number. Since this is tied to natural numbers, we could expect induction to be helpful here but I can't seem to find a relationship between consecutive prime numbers.

Answer 1

If we think modulo 6, we see that, for $n\geq 12$, only the numbers of the shape $6k+1$ and $6k-1$ can be prime. That is, at most 1/3 of the integers can be prime. For a given integer $m$, there are at most $\lfloor m/3\rfloor$ primes less than $m$, so the $\lfloor m/3\rfloor$-rd prime is at least a big as $m$. Some details are needed to polish this off. If more slack is needed in the inequality, we can think modulo 30, where less than 4/15-ths of the numbers are prime.