Using the prime number theorem I can show that $\displaystyle p_n / {\log{p_n}} \sim n$, where $\displaystyle p_n$ denotes the nth prime. I can also show that $\displaystyle x/{\log{x}}$ is an increasing function (for $\displaystyle x>e$). I'm having problems deducing that there are only finitely many $\displaystyle n$ such that $\displaystyle p_n > n (\log{n})^2$. I've tried showing that the limit (if it exists) of $\displaystyle p_n / {n (\log{n})^2}$ is strictly less than 1, but I can't get anywhere.